The tools we’ll use: divisors, line bundles, sheaves, cohomology, intersection theory, and Riemann-Roch. Almost all surfaces we consider in this class will be smooth. Let \(X\) be a smooth algebraic variety. A divisor \(D = \sum n_i Z_i\) with \(n_i\in {\mathbf{Z}}\) and \(Z_i\) codimension 1 subvarieties. On a curve, divisors are weighted sums of points, and on a surface they are weighted sums of curves. We say \(D_1\sim D_2\) if \(D_1 - D_2 = (\phi)\) with \(\phi\in {\mathbf{C}}(X)\) a rational function. These are principal divisors. We define \(\operatorname{Pic}(X) \mathrel{\vcenter{:}}=\operatorname{Div}(X)/\sim\).

Over \({\mathbf{C}}\), one has the GAGA principle: if \(X \subseteq {\mathbf{CP}}^n\) is a closed analytic subset, by the Chow lemma it is algebraic. One can then compute cohomology using analytic tools or algebraic tools.

For example, one can identify \(\operatorname{Pic}(X)\) as line bundles modulo isomorphism, i.e. \(H^1({\mathcal{O}}_X^{\times})\). Note that \({\mathcal{O}}_X(U)\) are regular functions, and \({\mathcal{O}}_X^{\times}(U)\) are nowhere vanishing regular functions – these are the first examples of sheaves.

Recall that a sheaf \(F\) of abelian groups is an assignment of open sets \(U\) to sections \(F(U)\) with appropriate restriction maps.

Recall that \({\mathcal{O}}_X(D)\) is locally isomorphic to \({\mathcal{O}}_X\), and sections of \({\mathcal{O}}_X(D)\) are rational functions \(\psi\) such that \((\psi) + D \geq 0\). Recall that \(D_1\sim D_2 \implies {\mathcal{O}}_X(D_1)\cong {\mathcal{O}}_X(D_2)\) by writing \(D_1-D_2 = (\zeta)\) and mapping \(s\mapsto s\zeta\). Checking \((\zeta) + D_1\geq 0\), one has \begin{align*} (\zeta \psi) + D_2 \geq 0 \iff (\zeta) + (\psi) + D_2 \geq 0 \iff D_1-D_2 + (\psi) + D_2 \geq 0 \iff D_1 + (\psi) + D_1 \geq 0 .\end{align*}

We say \({\mathcal{O}}_X(D)\) is locally free of rank 1, or invertible, or a line bundle.

Note that \(\mathop{\mathrm{Aut}}({\mathcal{O}}_X) = {\mathcal{O}}_X^{\times}\).

There is a SES \begin{align*} 0\to I\to {\mathcal{O}}_X \to {\mathcal{O}}_C \to 0 \end{align*} where \(I\cong {\mathcal{O}}_X(-C)\). These are examples of coherent sheaves, those which can locally be written \({\mathcal{O}}^{\oplus J}\to {\mathcal{O}}^{\oplus I}\to F\to 0\). An example of a non-coherent sheaf is \({\mathcal{O}}_X^{\times}\), since one can not multiply a section by a function with a zero. Other examples include locally constant sheaves \(\underline{{\mathbf{Z}}}, \underline{{\mathbf{C}}}\).

Given \(A\hookrightarrow B\twoheadrightarrow C\) there is a LES \(H^i(A)\to H^{i}(B)\to H^{i}(C)\to H^{i+1}(A)\to\cdots\). By Grothendieck, \(H^i(F) = 0\) for \(i > d = \dim X\) if \(F\) is coherent and \(X\) is a Noetherian topological space. If \(X\) is projective and \(F\) is coherent, \(H^i(F)\) all form finite-dimensional vector spaces. We write these dimensions as \(h^i(F)\), which are numerical invariants. Define \(\chi(F) = \sum_i (-1)^i h^i(F)\). One has \(\chi(B) = \chi(A) + \chi(C)\), since \(\chi\) of a LES is zero.

Recall that the Betti numbers are defined as \(\beta_i \mathrel{\vcenter{:}}=\dim_{\mathbf{Q}}H^i(X;{\mathbf{Q}})\) and \(\chi(X) \mathrel{\vcenter{:}}=\sum (-1)^i \beta_i\). Note that \(\chi(X) \neq \chi({\mathcal{O}}_X)\) in general: let \(C\) be a smooth projective curve of genus \(g\) over \({\mathbf{C}}\). Check that \(\chi(X) = 1 - 2g + 1 = 2-2g\), since there are \(2g\) generators for homology:

However, recall \(h^1 = h^{1, 0} + h^{0, 1} = g+g\), and \(\chi({\mathcal{O}}_X)= h^0({\mathcal{O}}_X) - h^1({\mathcal{O}}_X) = 1-g = {1\over 2}(2-2g)\) instead.

Recall the exponential exact sequence \(0\to {\mathbf{Z}}\xrightarrow{\cdot 2\pi i} {\mathcal{O}}_X \xrightarrow{\exp} {\mathcal{O}}_X^{\times}\to 1\). Taking the LES yields:

\begin{align*}\begin{tikzcd} {H^0({\mathbf{Z}}) = {\mathbf{Z}}} && {H^0({\mathcal{O}}_X) = {\mathbf{C}}} && {H^0({\mathcal{O}}_X^{\times}) = {\mathbf{C}}^{\times}} \\ \\ {H^1({\mathbf{Z}}) = {\mathbf{Z}}^{2g} \oplus \mathrm{tors}} && {H^1({\mathcal{O}}_X) = {\mathbf{C}}^g} && {H^1({\mathcal{O}}_X^{\times}) = \operatorname{Pic}(X)} \\ \\ {H^2({\mathbf{Z}}) = {\mathbf{Z}}^n\oplus\mathrm{tors}} && {H^2({\mathcal{O}}_X)} && \cdots \arrow[from=1-1, to=1-3] \arrow[from=1-3, to=1-5] \arrow[from=1-5, to=3-1] \arrow[from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow[from=3-5, to=5-1] \arrow[from=5-1, to=5-3] \arrow[from=5-3, to=5-5] \end{tikzcd}\end{align*} > Link to Diagram

We thus get a SES \(0\to {\mathbf{C}}^g/{\mathbf{Z}}^{2g} \to \operatorname{Pic}(X) \to F \to 0\) where \(F\) is some finitely generated abelian group. We define \(\operatorname{Pic}^0(X) = \operatorname{Jac}(X) = {\mathbf{C}}^g/{\mathbf{Z}}^{2g}\), which is an algebraic torus over \({\mathbf{C}}\) of dimension \(g\), and is the continuous part. We identify \(F \mathrel{\vcenter{:}}={\operatorname{NS}}(X)\) as the Néron-Severi group. Note that divisors in \(\operatorname{Pic}^0(X)\) are numerically zero, since intersection numbers are integers that vary continuously.

Note that \(D_1, D_2\in \operatorname{Pic}^0(X)\) means \(D_1\) is algebraically equivalent to \(D_2\), i.e. can be varied continuously. For linear equivalence, one uses \({\mathbf{P}}^1\) as the base, since rational functions on \(X\) are equivalent to maps to \({\mathbf{P}}^1\). Algebraic equivalence allows for the base to be an arbitrary curve.

Let \(X\) be a smooth projective variety, and recall \(\operatorname{Pic}(X) = \operatorname{Div}(X)/\sim\) or the group of rank 1 locally free sheaves \(F\) where \({ \left.{{F}} \right|_{{U}} } = { \left.{{{\mathcal{O}}_X}} \right|_{{U}} }\). Both are isomorphic to \(H^1({\mathcal{O}}_X^{\times}) \cong \check{H}^1({\mathcal{O}}_X^{\times})\), the ech cohomology. Let \({\mathcal{U}}\) be an open cover of \(X\), so \(X = \cup_i U_i\). Then a divisor \(D\) on \(U_i\) locally has an equation \(f_i\) for some rational function, i.e. \((f_i) = D\). Consider \(\left\{{f_i \in K^{\times}(U_i)}\right\}\). Then on \(U_{ij}\), write \(g_{ij} = {f_i \over f_j}\). This has no zeros and no poles, so \((g_{ij}) = 0\). Thus \(g_{ij}\) and \(1/g_{ij}\) are both regular and give sections \(g_{ij}\in {\mathcal{O}}^{\times}(U_{ij})\). We get a collection \(\left\{{g_{ij} \in {\mathcal{O}}^{\times}(U_{ij}) {~\mathrel{\Big\vert}~}g_{ij}\cdot g_{jk} \cdot g_{ki} = 1 \text{ on } U_{ijk}}\right\}\) modulo \(f_i' = f_i g_i\) for \(g_i\in {\mathcal{O}}^{\times}(U_i)\). This is precisely \(\check{C}^1/\check{B}^1 = \check{H}^1({\mathcal{U}}, {\mathcal{O}}_X^{\times})\). One defines \(\check{H^1}(X, F) = \mathop{\mathrm{colim}}_{{\mathcal{U}}} \check{H}^1({\mathcal{U}}, F)\) for a sheaf \(F\).

For any projective variety, there are three important sheaves:

• \({\mathcal{O}}_X\),
• \({ \left.{{ {\mathcal{O}}_{{\mathbf{P}}^n}(1) }} \right|_{{X}} } = {\mathcal{O}}_X(1)\) for \(X \hookrightarrow{\mathbf{P}}^N\) a projective embedding,
• \(\omega_X = {\mathcal{O}}_X(K_X)\).

On a normal variety, \(\operatorname{CDiv}(X) \hookrightarrow\operatorname{Div}(X)\) is a subgroup. There is always such a map, even for non-normal varieties, but generally \(\operatorname{CDiv}(X)\) is bigger.

Letting \({\mathcal{U}}\) be an open cover, consider \(U_{ij}\). We have \({ \left.{{F}} \right|_{{U_i}} }\cong {\mathcal{O}}_{U_i}\), and similarly we get a diagram

\begin{align*}\begin{tikzcd} {{ \left.{{F}} \right|_{{U_{ij}}} }} && {{\mathcal{O}}_{U_{ij}}} & {g_{ij}\in {\mathcal{O}}^{\times}(U_{ij})} \\ \\ {{ \left.{{F}} \right|_{{U_{ij}}} }} && {{\mathcal{O}}_{U_{ij}}} & 1 \arrow["{\alpha_i}"', hook, two heads, from=1-1, to=1-3] \arrow[Rightarrow, no head, from=1-1, to=3-1] \arrow["{\alpha_j}", hook, two heads, from=3-1, to=3-3] \arrow[from=3-3, to=1-3] \arrow[maps to, from=3-4, to=1-4] \end{tikzcd}\end{align*} > Link to Diagram

If \(\dim X = n\), then \(\omega_X \mathrel{\vcenter{:}}=\Omega_{X}^n\) which has sections \(\Omega_X(U) = \left\{{f dx_1 \wedge \cdots \wedge dx_n }\right\}\) where \(f\) is regular. This is an invertible rank 1 locally sheaf. We have \(\Omega_X^1 = \left\{{f_1 dx_1 + \cdots + f_n dx_n}\right\}\) which is rank \(n\), locally isomorphic to \({\mathcal{O}}_X^{\oplus n}\) since the \(dx_i\) form a basis. Similarly we have \(\operatorname{rank}\Omega_X^k = {n\choose k}\).

To compute for other varieties, we do this indirectly using the adjunction and Hurwitz formula. For \(Y \subseteq X\), we have \(K_Y = (K_X + Y)\mathrel{\Big|}_Y\), or \(\omega_Y = (\omega_X \otimes{\mathcal{O}}_X(Y))\mathrel{\Big|}_Y\). For \(C \hookrightarrow{\mathbf{P}}^2\) a curve of degree \(d\), this yields \(K_C = (d-3)H\mathrel{\Big|}_C\). One checks \(2g-2 = \deg K_C = d(d-3)\), which implies \(g = {1\over 2}(d-1)(d-2)\).

Consider \(S_d \hookrightarrow{\mathbf{P}}^3\) a hypersurface of degree \(d\) cut out by some \(f_d\). Then \(K_S = (d-4)H\mathrel{\Big|}_S\), and \(d=4\implies K_S = 0\) and this gives a K3 surface. Similarly \(d<4 \implies K_S > 0\) and \(d<4\implies K_S < 0\).

The Hurwitz formula: let \(\pi: X\to Y\) be a map of curves, then \(K_X = \pi^* K_Y + R\) where \(R = \sum(n_i - 1)p_i\) where \(n_i\) is the ramification number – given in local equations as \(y=x^{n_i}\). Note that if \(y=x^n\) then \(dy = nx^{n-1} dx\). This equation generalizes to ramified maps of surfaces with the same formula.

Generally for a 2-to-1 cover \(\pi: S\to {\mathbf{P}}^2\), one has \(R = {1\over 2}\pi^*B\) and \begin{align*} K_S = \pi^* K_{{\mathbf{P}}^2} + R = \pi^*(K_{{\mathbf{P}}^2} + {1\over 2} B)= \pi^* (-3 + d/2)H .\end{align*}

Recall that \(h^{i}(D) = h^{n-i}(K-D)\) by Serre duality. Define

• \(q\mathrel{\vcenter{:}}= h^1({\mathcal{O}})\) the irregularity,
• \(p_g \mathrel{\vcenter{:}}= h^2({\mathcal{O}})\) the geometric genus,
• \(h^1(\omega) = h^1({\mathcal{O}}) = q\),
• \(h^2(\omega) = h^0({\mathcal{O}}) = 1\),
• \(h^0(\omega) = h^2({\mathcal{O}}) = p_g\),
• \(h^0(mK) = p_m\) the plurigenera,
• Consider \(\lim_{m\to \infty} p_m \sim m^{\kappa}\), then \(\kappa\) is the Kodaira dimension.

There is a symmetric pairing \({\operatorname{NS}}(X)\times {\operatorname{NS}}(X)\to {\mathbf{Z}}\) given by intersecting curves and counting points. Recall there was a SES \begin{align*} 0\to {H^1({\mathcal{O}}_X) \over H^1(X;{\mathbf{Z}})}\to \operatorname{Pic}(X) \to {\operatorname{NS}}(X) \mathrel{\vcenter{:}}=\ker\qty{H^2(X;{\mathbf{Z}})\to H^2({\mathcal{O}}_X)}\to 0 .\end{align*} We have \({\operatorname{NS}}(X)/{\operatorname{tors}}\cong{\mathbf{Z}}^{\rho}\) where \(\rho\) is the Picard rank. Note that this is compatible with the cup product pairing when working with varieties over \({\mathbf{C}}\). A definition that works over any field: \(C_1\cdot C_2 = \sum_{p\in C_1 \cap C_2} \mu_p(C_1, C_2)\) where \(\mu_p(C_1, C_2) = \dim_{\mathbf{C}}{\mathcal{O}}_{S, p}/\left\langle{f, g}\right\rangle\) where \(f,g\) are local equations for the \(C_i\) near \(p\). This works when \({\sharp}(C_1 \cap C_2)< \infty\).

The moving lemma: \(\forall C_2 = A-B\) where \(A, B\) are general hyperplanes for some \(S \hookrightarrow{\mathbf{P}}^{n_i}\). Note that if \(C\hookrightarrow S\) is smooth, then \(C^2 = \deg N_{C/S}\), the degree of the normal bundle.

Recall that \({\operatorname{NS}}(X)\) carries an integral intersection form. For curves without common components, it counts the intersection points with multiplicities.

For all \(f:X\to Y\) maps of varieties, define \(f^*: \operatorname{Pic}(Y)\to \operatorname{Pic}(X)\) and \({\operatorname{NS}}(Y)\to {\operatorname{NS}}(X)\), defined by pulling back equations. Define \(f_*: \operatorname{Pic}(X)\to \operatorname{Pic}(Y)\) by \begin{align*} f_* C = \begin{cases} 0 & f(C) = {\operatorname{pt}}\\ dC' & f(C) = C', d=\deg(C\to C') \end{cases} .\end{align*}

Some properties:

1. If \(f: S'\to S\) is generically degree \(d\) then \begin{align*} (f^* D_1)\cdot (f^* D_2) = d D_1 \cdot D_2 .\end{align*}

2. Projection formula: \begin{align*} f^* D_1 \cdot D_2 = D_1 \cdot f_* D_2 .\end{align*}

3. Hirzebruch-Riemann-Roch: let \(X\) be a smooth projective of dimension \(n\) and \(L = {\mathcal{O}}_X(D)\) a line bundle on \(X\). Then \begin{align*} \chi(L) = \int_X e^D \mathrm{Td}_X .\end{align*} Here \(e^D = \sum_k D^k/k!\) and \(\mathrm{Td}_X = 1 + {1\over 2}c_1 + {1\over 12}(c_1^2 + c_2) + \cdots\) and \(c_i\) are the Chern numbers of \(T_X\).

Recall that \(c_1: \operatorname{Pic}(X)\to H^2(X; {\mathbf{Z}})\). Note that \(c_1 = -K_X\) and \(c_n = \chi_{{\mathsf{Top}}}(X)\). Here \(\int_X({-})\) is the degree of a cycle of codimension \(n\), hence dimension 0, and counts its number of points.

Recall that for a curve \(D\) on a surface \(S\), one has \(\deg K_D = 2p_a(D) - 2\) and \(K_D = \qty{K_S + D}\mathrel{\Big|}_D\). From the genus formula, \(p_a(D) = {D(D+K)\over 2} + 1\).

Note that \(\chi_{\mathsf{Top}}({\mathcal{O}}_S(D))\) can be computed easily for toric varieties. The Euler characteristic is additive, and every such variety decomposes as \(({\mathbf{C}}^{\times})^2 {\textstyle\coprod}{\mathbf{C}}^{\times}{\textstyle\coprod}\left\{{p_i}\right\}\), and \(\chi({\mathbf{C}}^{\times}) = 0\), so only points \(p_i\) contribute.

Recall that rational maps on irreducible varieties are regular functions on open subsets, and correspond to maps \(X\to {\mathbf{A}}^1\). Note that any two open subsets of an irreducible variety intersect. One can add, multiply, and invert nonzero rational functions by throwing out closed sets of zeros, so these form a field. Say \(X\overset{\sim}{\dashrightarrow}Y\) are birationally isomorphic if they share a common open set \(X \supseteq U \subseteq V\). Say \(X\) is rational iff \(X\overset{\sim}{\dashrightarrow}{\mathbf{P}}^n\) or \({\mathbf{A}}^n\), and \(X\) is unirational if there exists a dominant rational map \({\mathbf{P}}^n\dashrightarrow X\) (covered by a rational variety). Dominant morphisms: closure of the image is the entire space. Note that dominant rational maps \(X\to Y\) yields an embedding of fields \({\mathbf{C}}(Y) \hookrightarrow{\mathbf{C}}(X)\). Thus rational varieties of dimension \(n\) satisfy \({\mathbf{C}}(X) = {\mathbf{C}}(x_1,\cdots, x_n)\), while unirational means \({\mathbf{C}}(X)\) is a subfield.

Luroth problem: does unirational imply rational? True for \(n=1,2\), we will prove that \(n=2\) case in this course. Generally false for \(n\geq 3\).

We will define a blowup of a smooth surface at a smooth point \(\pi: \operatorname{Bl}_p S\to S\). We start with the example of \(S={\mathbf{A}}^2\) and \(p\) an arbitrary point. This will be birational because \({\mathbf{A}}^2\setminus\left\{{p}\right\} \cong \operatorname{Bl}_p{\mathbf{A}}^2\setminus E\). Different lines passing through \(p\) will become disjoint curves intersecting \(E\).

In equations, write \(\operatorname{Bl}_p {\mathbf{A}}^2 = \left\{{xY=yX}\right\} \subseteq {\mathbf{A}}^2 \times {\mathbf{P}}^1\) with coordinates \(x,y\) and \(X,Y\) respectively. Note that \(x/y = X/Y\) if \(y,Y\neq 0\).

In general, for \(\tilde S\to S\), let \(x,y\) be local parameters for \(p\in S\). One takes \({\mathcal{O}}_{S, p}\supseteq{\mathfrak{m}}_p = \left\langle{x,y}\right\rangle\) with \(T_p {}^{ \vee }X = \left\langle{dx, dy}\right\rangle\) and \(\widehat{{\mathcal{O}}_{S, p}} = {\mathbf{C}}{\llbracket x,y \rrbracket }\). So any local regular function \(s\) can be approximated by a power series.

Claim: \(E^2 = -1\). Write \({\mathbf{P}}^1 = {\mathbf{A}}^1\cup{\mathbf{A}}^1\) with \(u=X/Y\) and \(v=Y/X\). Write \(\operatorname{Bl}_p {\mathbf{A}}^2 = {\mathbf{A}}^2_{u, y} \cup{\mathbf{A}}^2_{v, x}\). This is a change of coordinates

• \(u,y = {x\over y}, y\) where \(x=uy\),
• \(v,x = {y\over x}, x\) where \(y=vx\).

If \(C\) is a line through \(p\in {\mathbf{A}}^2\), which can be replaced with a curve \(C\) that is smooth at \(p\), then \(f^*C = C' + E\) where \(C' = \overline{f^{-1}(C \setminus p)}\) is the strict transform. Write \(C = \left\{{g=0}\right\} = \left\{{y+\alpha x = 0}\right\}\). Then \(f^* g = vx + ax = x(v+a)\) where \(x=0\) is the equation of \(E\) and \(v+a\) is the equation of \(C'\). Conclude using the projection formula: \(f^* C \cdot E = C \cdot f_* E = C\cdot 0\) on one hand, and is equal to \((C'+E)E = C'E + E^2 = 1+E^2\) on the other hand, so \(E^2 = -1\).

Recall the construction of the blowup at a point \(\pi: \tilde S\to S\). We proved \(E^2= -1\).

Blow up the nodal curve to get \(p_a' = p_a - 1\). Blow up the cuspidal curve to get….something.

Note that all cubics have the same \(p_a\), since it only depends on the linear equivalence class of \(C\).

Consider resolving \(C\) to \(C'\) by \(\nu\) and use the fundamental SES \begin{align*} 0\to {\mathcal{O}}_C \to \nu^* {\mathcal{O}}_{\tilde C} \to F\to 0 .\end{align*} Then \(\chi({\mathcal{O}}_{\tilde C}) = \chi({\mathcal{O}}_C) + \chi(F)\) where \(\chi(F) = h^0(F) = d > 0\). Recall \(p_a = 1 - \chi({\mathcal{O}}_C)\). If \(C\) is smooth then \(p_a(C) = g\).

On linear systems: let \(V \subseteq H^0(L)\) where \(L = {\mathcal{O}}(C)\). Let \(s\in V\), then \((s) = D\) an effective divisor. Write \({\mathbf{P}}V = \left\{{ (s) {~\mathrel{\Big\vert}~}s\neq 0}\right\}\), this is equivalent to the set of effective divisors \(D\sim C\). Recall \(|C| = H^0({\mathcal{O}}(C)) = \left\{{ \phi \text{ rational } {~\mathrel{\Big\vert}~}( \varphi) + C = D \geq 0}\right\}\), i.e. those \((\phi)\) such that \((\phi) + C\) has no poles. This is a complete linear system. Note that \(( \varphi_1) = (\varphi_2) \iff \qty{ \varphi_1\over \varphi_2} = 0\). Here \(s\) and \(\varphi\) are in bijection.

Let \(f: S \to {\mathbf{P}}^n\) such that \(f(S)\) is not contained in a hyperplane. This yields a linear system. In projective coordinates \(x_i\) of \({\mathbf{P}}^n\), write a hyperplane \(H\) as \(\sum c_i x_i \in {\mathbf{P}}V\) where \({\mathbf{P}}^n = {\mathbf{P}}V {}^{ \vee }\). Then \(f^* H\) is a divisor on \(S\). Define a line bundle \(L = f^* {\mathcal{O}}_{{\mathbf{P}}^n}(1)\) on \(S\). Each \(H\) defines a section \(s\). This yields an \((n+1)\)-dimensional vector space \(V \subseteq L\).

Write the base locus of \({\mathbf{P}}V\) as \(\bigcap_{D\in {\mathbf{P}}V} D\). A base component is a divisor in the base locus. Say \({\mathbf{P}}V\) is basepoint free if \(\mathrm{Base}({\mathbf{P}}V) = \emptyset\). We then get a bijection between linear systems without base components for smooth varieties \(S\) and rational maps \(S\dashrightarrow{\mathbf{P}}^n\).

Write the uniformizer of \({\mathcal{O}}_{C, p}\) as \(t\), let \(\phi_0, \cdots, \phi_n\) be rational functions. Write \(\phi_i = u t^{m_i}\), take \(d=\min m_i\), and divide by \(t^d\).

Next time:

• \(f\) is injective if \(L\) separates points: \(\forall p,q, \exists D\) such that \(p\in D\) but \(q\not\in D\).
• \(f\) is a closed embedding if \(L\) separates tangent vectors.

This prevents the following situation, where a tangent vector is sent to zero:

Recall that a linear system is defined by \(L = {\mathbf{P}}(V)\) where \(V \subseteq H^0(F)\) is a finitely-generated vector space. We define \({ \operatorname{Base}}(L) = \cap(s)\) for \(s\in V\). Recall that if \(F\) is a line bundle then \(F = {\mathcal{O}}_X((s))\) for any effective \((s)\).

Recall that

• \(f\) is injective \(\iff L\) separates points.
• \(f\) is an isomorphism \(\iff\) additionally \(L\) separates tangent directions.

Writing \({\mathbf{T}}_{X, p} = ({\mathfrak{m}}_p/{\mathfrak{m}}_p^2) {}^{ \vee }\), the latter condition means \(T_p \to T_{f(p)}\) is injective, or equivalently \(T {}^{ \vee }_{f(p)}\to T {}^{ \vee }_{p}\) surjective. This is satisfied if the pullbacks \(f^* y_1,\cdots, f^* y_n\) generate \({\mathfrak{m}}_p/{\mathfrak{m}}_p^2\), where \(y_1,\cdots, y_n\) are local coordinates in \({\mathbf{P}}^n\). Note that a surjection on maximal ideals corresponds to a surjection on the completions of local rings \(\widehat{{\mathcal{O}}_{f(p)}}\twoheadrightarrow\widehat{{\mathcal{O}}_p}\).

For \(X\) a smooth curve, these conditions simplify:

• \({ \operatorname{Base}}({\left\lvert {D} \right\rvert}) = \emptyset\), or equivalently \(h^0(D-p) = h^0(D)-1\). These numbers are equal iff every divisor passes through \(p\). This is proved by Riemann-Roch.
• Separating points: \(h^0(D-p-q) = h^0(D-p)-1\), so not all of the divisors that pass through \(p\) also pass through \(q\).
• Separating tangents: \(h^0(D-2p) = h^0(D-p)-1\).

Note that on a surface, \({\left\lvert {D-p} \right\rvert}\) is a just convenient notation for divisors that pass through \(p\), since \(D-p\) is not a divisor.

One can always remove the indeterminacy in codimension 1, but not necessarily in codimension 2.

Idea:

On the proof: let \(D = D_0\) be a divisor in the linear system. Resolve to get \(D_1 = f_1^* D - m_1 E_1\) where \(m_1 = {\operatorname{mult}}_p(L) \mathrel{\vcenter{:}}=\min \left\{{{\operatorname{mult}}_p(D) {~\mathrel{\Big\vert}~}D\in \L}\right\}\). Check that \(D_1^2 = D_0^2 - m_1^2\) and \(D_i^2\geq 0\) for all \(i\). But the self-intersection can’t decrease forever.

Let \(L = {\mathbf{P}}^1\) in \({\mathbf{P}}^2\) be the set of lines through zero. Blowing up yields a fibration \(\operatorname{Bl}_p {\mathbf{P}}^2 \to {\mathbf{P}}^1\) which is a ruled surface with fibers the exceptional curves. This surface is \({ \mathbf{F} }_1\).

Blowup enough to make the rational map regular. Either it’s an isomorphism or undefined at a point. In the latter case, blowup and apply the theorem. Why this terminates: \(f^{-1}(p)\) contains finitely many divisors.

Note that this proof does not use Castelnuovo’s criterion.

Last time: \(S' \xrightarrow{f} S\) with \(S\) smooth and \(f^{-1}\) undefined at \(p\in S\). Then \(f\) contracts a nonempty curve \(C \subseteq S'\), i.e. \(f(C) = p\).

Recall Zariski’s main theorem: if \(f:X\to Y\) is finite and birational with \(Y\) normal, then \(f\) is an isomorphism. Recall that \(Y\) is normal iff \({\mathcal{O}}_{Y, p}\) is integrally closed in its fraction field, and smooth implies normal because UFDs are integrally closed. We give two proofs of the theorem from last time.

Slogan: non-regular rational maps insert curves. Note that it may factor as a sequence of blowups. Recall Castelnuovo’s contractibility criterion: rational \((-1)\)-curves can be blown down.

Normalization produces a Stein factorization:

\begin{align*}\begin{tikzcd} & {\tilde S'} \\ \\ S && {S'} & {{\mathbf{P}}^n} \arrow[from=1-2, to=3-3] \arrow[from=3-1, to=1-2] \arrow[from=3-1, to=3-3] \arrow[hook, from=3-3, to=3-4] \end{tikzcd}\end{align*} > Link to Diagram

Here \(\tilde S'\) is the normalization, and can be written as \(\tilde S' = \operatorname{Spec}f_* {\mathcal{O}}_S\).

Grothendieck’s theorem on formal functions: let \(f:X\to Y\) be projective, \(F\in {\mathsf{Coh}}(X)\), and \(p\in Y\). Consider the higher direct images \({\mathbf{R}}^i f_* F\) on \(Y\), which is an \({\mathcal{O}}_Y\)-module. Complete the stalk at \(p\) to get \(\widehat{{\mathbf{R}}^i f_* F}_p\) which is a module over \(\widehat{{\mathcal{O}}_{Y, p}}\). This can be computed as a limit \(\mathop{\mathrm{colim}}_{Z} F\mathrel{\Big|}_Z\) where \(Z\) are thickenings of the fiber \(f^{-1}(p)\).

We first apply this for \(i=0\); we get \(\widehat{f_* {\mathcal{O}}_S} =\widehat{{\mathcal{O}}_{S'}}\). For smoothness, we want to show the latter is isomorphic to \({\mathbf{C}}{\llbracket x,y \rrbracket }\). The theorem says to compute \(\mathop{\mathrm{colim}}_m H^0(mE)\). We have \(0\to {\mathcal{O}}_S(mE)\to {\mathcal{O}}_S \to {\mathcal{O}}_{mE}\). The claim is that the colimit is \({\mathbf{C}}[x,y,x^2,xy,y^2,\cdots,y^m]\), the monomials up to the \(m\)th order. Use the fact that \({\mathcal{O}}_E(-mE) = {\mathcal{O}}_{{\mathbf{P}}^1}(m)\).

This concludes chapter 2. Next time: ruled surfaces.

Recalling ruled surfaces: let \(f:S\to C\) where all fibers are isomorphic to \({\mathbf{P}}^1\). We proved it is a locally free \({\mathbf{P}}^1\) bundle, so \(\forall p\in C\) there is some \(U\ni p\) such that \(f^{-1}(U) \cong U\times {\mathbf{P}}^1\).

Classification of locally free rank \(r\) sheaves \({\mathcal{E}}\) and locally free \({\mathbf{A}}^1\) bundles (vector bundles). Letting \(X = \bigcup U_i\), there is an isomorphism \(g_i: { \left.{{{\mathcal{E}}}} \right|_{{U_i}} } { \, \xrightarrow{\sim}\, }{\mathcal{O}}_{U_i}^{\oplus r}\). Different choices of isomorphisms yield elements \(g_i\in \operatorname{GL}_r({\mathcal{O}}(U_i)) = \mathop{\mathrm{Aut}}({\mathcal{O}}_{U_i}^{\oplus r})\). The coefficients are in the ring of regular functions on \(U_i\). How do these isomorphisms glue? We specify transition functions \(g_{ij}\in \operatorname{GL}_r({\mathcal{O}}(U_i \cap U_j))\) satisfying the 1-cocycle condition on triple intersections \(g_{ij}g_{jk}g_{ki} = 1 \in \operatorname{GL}_r({\mathcal{O}}(U_i \cap U_j \cap U_k))\). We mod out by the relation \(g_{ij}' = g_j^{-1}g_{ij} g_i\), the 1-coboundary condition. This yields Čech cohomology \(\check{H}^1(U, \operatorname{GL}_r({\mathcal{O}})) \cong H^1(X, \operatorname{GL}_r({\mathcal{O}}))\). This works generally: global versions on locally free objects live in \(\check{H}^1(U, \mathop{\mathrm{Aut}}({\mathcal{E}}))\).

Vector bundles are locally free \({\mathbf{A}}^r\) bundles. Let \(f:Y\to X\) have fibers \({\mathbf{A}}^1\), so \(U_i \subseteq X\) lifts to \(f^{-1}(U_i) = U_i \times {\mathbf{A}}^r\). What are the transition functions? One looks for automorphisms of \((U_{i} \cap U_j) \times {\mathbf{A}}^r\ni (x,y_1,\cdots, y_r)\), which correspond to \(g_{ij}(x)\in \operatorname{GL}_r({\mathcal{O}}(U_i \cap U_j))\). What is the corresponding sheaf? Let \({\mathcal{E}}(U)\) be sections over \(U\), so morphisms \(s: U\to f^{-1}(U)\). Locally this is given by \(r\) regular functions, so \({\mathcal{E}}(U_i) = {\mathcal{O}}(U_i)^{\oplus r}\).

For ruled surfaces, we consider \(\check{H}^1(C, \operatorname{PGL}_2({\mathcal{O}}))\). More generally, take \(\operatorname{PGL}_r({\mathcal{O}})\).

Thus \({\mathcal{E}}\) on \(C\) is in bijection with an \({\mathbf{A}}^2\) bundle \({\mathbf{A}}({\mathcal{E}})\to C\). What is \({\mathbf{P}}_C({\mathcal{E}})\)? This is the bundle of lines in the vector bundle. Dually, one could define this as rank 1 quotients of the vector bundle. Passing back and forth: replace \({\mathcal{E}}\) by \({\mathcal{E}} {}^{ \vee }\).

Goal: understand locally free sheaves on curves, e.g. \({\mathbf{P}}^1\) (easy), elliptic curves (harder), and curves of general type (generally hard). Next: relate birationally and geometrically ruled surfaces.

A (classically) minimal surface is a smooth projective surface with no \(-1\) curves.

Ruled surface: \(S\overset{\sim}{\dashrightarrow}C\times {\mathbf{P}}^1\) for \(C\) a curve. Take a sequence of blowups to obtain \(\tilde S\to C\times {\mathbf{P}}^1\) a regular morphism. Consider the composite \(\tilde S\to C\times {\mathbf{P}}^1\to C\); most fibers are \({\mathbf{P}}^1\) but some are more complicated:

Claim: either \(\tilde S = {\mathbf{P}}_C({\mathcal{E}})\) or there exists a \(-1\) curve in a fiber. In the first case, \(\tilde S\to {\mathbf{P}}_C({\mathcal{E}})\to C\) is a relatively minimal model.

Fact: there are no covers \({\mathbf{P}}^1\to C\) if \(g(C) > 0\). By Riemann-Hurwitz, \(K_{{\mathbf{P}}^1} = f^* K_C + \sum (m_i - 1)p_i\), and taking degrees yields \(-2\) on the LHS and \(2g-2 > 0\) and a positive contribution on the RHS, a contradiction.

Alternatively, consider \(\pi: C'\to C\) and consider the induced map \(\pi^*: H^0(\Omega_C)\to H^0(\Omega_{C'})\). Note \(y=f(x) \implies dy = f'(x) dx\). The claim is that this map is injective, so the genus increases.

Note \(\operatorname{Ext}(A, B) = \operatorname{Ext}({\mathcal{O}}, B \otimes A^{-1}) = H^1(B \otimes A^{-1})\). Considering an elliptic curve and \(0\to {\mathcal{O}}\to {\mathcal{E}}\to {\mathcal{O}}\to 0\), one has \(H^1({\mathcal{O}}\otimes{\mathcal{O}}^{-1}) = H^1({\mathcal{O}}) = g(C) = 1\), so there is a nontrivial extension.

Next time: \(\deg {\mathcal{E}}= \deg \operatorname{det}{\mathcal{E}}\) as a line bundle.

We proved that every geometrically ruled surface is of the form \({\mathbf{P}}_C({\mathcal{E}})\) with \(\operatorname{rank}{\mathcal{E}}= 2\). Recall that locally free \({\mathcal{O}}_C\) modules are in bijection with \({\mathbf{A}}^n\) bundles over \(C\).

Recall Riemann-Roch for line bundles on a curve: \(\chi(L) = \deg L + 1 - g\). Note that \(\operatorname{det}({\mathcal{E}}) \cong L\otimes M\), so \(\deg({\mathcal{E}}) = \deg(L) + \deg(M)\). Thus \begin{align*} \chi({\mathcal{E}}) = \chi(L) + \chi(M) = (\deg L + 1-g) + (\deg M + 1-g) = \deg({\mathcal{E}}) = 2(1-g) .\end{align*}

When does this extension split? Twist \(L\to {\mathcal{E}}\to M\) by \(M^{-1}\) to get \(L\otimes M^{-1}\to {\mathcal{E}}\otimes M^{-1}\to {\mathcal{O}}\). A section of the former is the same as a map \({\mathcal{O}}\to {\mathcal{E}}\otimes M^{-1}\). A section \(s\in H^0({\mathcal{E}}\otimes M^{-1})\) maps to \(1\in H^0({\mathcal{O}})\) which further maps to some \(e\in H^1(L\otimes M^{-1})\) in the associated long exact sequence. The extension splits iff the extension class \(e=0\). As a corollary, if \(H^1(L\otimes M^{-1}) = 0\) then the extension splits.

On \({\mathbf{P}}^1\), any rank \(r\) vector bundle \(E\) splits as a sum of \({\mathcal{O}}(n_i)\).

Why this decomposition is unique: this minimum of \(n_1, n_2\) is unique as a function of \(E\). The \(\max(n_1, n_2)\) is the maximal \(n\) such that \(h^0(E(-n)) > 0\), since twisting by \(-n\) yields \({\mathcal{O}}\oplus {\mathcal{O}}(n_2-n)\) where \(n_2 - n \leq 0\).

For an elliptic curve \(C\), the only indecomposable rank 2 bundles are extensions \(E\) in \({\mathcal{O}}\to E\to {\mathcal{O}}\) where \(e\in H^1({\mathcal{O}}) = {\mathbf{C}}\), or \({\mathcal{O}}\to E\to {\mathcal{O}}(p)\) with \(e\in H^1({\mathcal{O}}(-p)) = H^0({\mathcal{O}}(p)) {}^{ \vee }= {\mathbf{C}}\) for \(p\in C\) a point. One can take either of these and tensor by a line bundle.

For curves of genus \(g\geq 2\), there is a moduli space of indecomposable rank 2 vector bundles of degrees 0 or 1. The dimension is \(3g-3\).

On \(S\), there is a natural SES \(N\to p^* E\to {\mathcal{O}}_{{\mathbf{P}}(E)}(1)\) where \(p: S\to C\), \(p^* E\) is rank 2, and \(N, {\mathcal{O}}_{{\mathbf{P}}(E)}(1)\) are line bundles. There is a similar sequence on fibers \({\mathcal{O}}(-1)\to {\mathcal{O}}\oplus {\mathcal{O}}\to {\mathcal{O}}(1)\). Regard a point in a fiber as an \({\mathbf{A}}^1 \subseteq {\mathbf{A}}^2\). We thus call \({\mathcal{O}}_{{\mathbf{P}}(E)}(1)\) the tautological bundle.

Sections \(s: C\to {\mathbf{P}}_C(E)\) are in bijection with rank 1 locally free quotients of \(E\) on \(C\).

Intersection theory on a deeper level, toward the enumerative geometry of \({\mathbf{P}}_C(E)\). Let \(X\) be a variety, and \(F\) a rank \(r\) vector bundle on \(X\). Then there exist Chern classes \(c_i(F)\in A^i(X)\) for \(0\leq i \leq \dim X\). Recall that \(A^i(X)\) are codimension \(i\) cycles \(\sum n_k Z_k\) modulo linear equivalence, so \(A^1(X) = \operatorname{Pic}(X)\). Recall linear equivalence:

The total Chern class is \(c(F) \mathrel{\vcenter{:}}=\sum_i c_i(F)\). It is additive in SESs: for \(A\to B\to C\), one has \(c(B) = c(A)\cdot c(C)\). A map \(f: X\to Y\) induces \(f^*: A^i(Y) \to A^i(X)\) where \(c_i(f^* F) = f^*(c_i(F))\). If \(L\) is a line bundle, then \(c(L) = 1 + L\).

By general principles, \(c_2(p^* E) = 0\). Next time \([N] \cdot [{\mathbf{P}}_C(E)] = 0\).

Let \(C\) be a curve of genus \(g\) and \(E\) a rank 2 vector bundle, and let \(S = {\mathbf{P}}_C(E)\to C\). There is an exact sequence \(0\to N\to p_* E\to {\mathcal{O}}_S(1)\to 0\) over \(S\) where the ranks are 1,2,1 respectively. Let \(h = c_1({\mathcal{O}}_S(1))\in {\operatorname{NS}}(S)\); on any fiber \(f\) this restricts to \({\mathcal{O}}_{{\mathbf{P}}^1}(1)\). Note that \(h\) need not be effective.

Note that twisting \(E\) by a line bundle changes \(h\) but does not change \({\mathbf{P}}_C(E)\).

Recall that \(N\) was defined by varying lines in \({\mathbf{A}}^2\) along a fiber. Twisting by a line bundle yields \(0\to N\otimes p^* L \to p^*(E\otimes L) \to {\mathcal{O}}_S(1)\otimes p^* L \to 0\). This yields \(h' \mathrel{\vcenter{:}}= h + p^* L\) and \((h')^2 = h^2 + 2\deg(L)\).

Basic numerical invariants:

• \(q(S) = h^1({\mathcal{O}}_S) = h^0(\Omega_S)\),
• \(p_g(S) = h^2({\mathcal{O}}_S) = h^0(\Omega_S^2)\),
• \(P_m(S) = h^0({\mathcal{O}}_S(mK)) = h^0( (\Omega_S^2)^{\otimes m} )\).

Note that \(P_1 = p_g\) and \(P_0 = 1\).

Rational maps thus yield inequalities in both directions.

We now consider ruled surfaces \(S \overset{\sim}{\dashrightarrow}C\times {\mathbf{P}}^1\). We obtain

• \(q(S) = g\),
• \(p_g(S) = 0\),
• \(P_m(S) = 0\).

Let \(p_i: C\times {\mathbf{P}}^1\to C, {\mathbf{P}}^1\) be the coordinate projections. Then \({\mathbf{T}}_{C\times {\mathbf{P}}^1} = p^* {\mathbf{T}}_{C} \oplus p^* T_{{\mathbf{P}}^1}\). Take duals to get \(\Omega_S = p^* \Omega_C \oplus \Omega_{{\mathbf{P}}^1}\) and check \(h^0(\Omega_S) = h^0(\Omega_C) + h^0(\Omega_{{\mathbf{P}}^1}) = g+0 = g\). Note that \(p_g(S) = h^0(K_S)\). Also note that \(K^2\) is not a birational invariant, since blowing up decreases this by 1. Here we have \(K_S^2 = 8-8g\) by computing \((-2h + (\deg E + 2g-2))^2 = 4\deg E - 4(\deg E + 2g-2)\). For \(g=0\), we get \({\mathbf{P}}^1\times {\mathbf{P}}^1\), check \(K_S^2 = (2h + 2f)^2 = 8\). For \(g=1\), check \(K_S^2 = 0\).

Note that a surface \(S\) is birationally ruled iff \(P_m(S) = 0\) for all \(m\). This is a classical theorem. It is rational iff \(q(S) = P_2(S) = 0\). Recall that Kodaira dimension is defined by \(P_m \sim m^{\kappa(S)}\). For \(\kappa(S) = 0\), there is a full classification in terms of the above invariants. There is a structure theorem for \(\kappa(S) = 1\), elliptic surfaces. For \(\kappa(S) = 2\), there are finitely many types where \(q=p_g=0\), but there is not a full classification.

New chapter: rational surfaces. Recall that a variety \(X\) is rational iff \(\exists f: X\overset{\sim}{\dashrightarrow}{\mathbf{P}}^N\) for some \(N\). Examples include Hirzebruch surfaces \({ \mathbf{F} }_n\), geometrically ruled surfaces over \({\mathbf{P}}^1\). These are fibrations \(S\to {\mathbf{P}}^1\) where every fiber is \({\mathbf{P}}^1\). Recall that geometrically ruled surfaces are of the form \({\mathbf{P}}_{{\mathbf{P}}^1}(V)\) where \(V\) is a vector bundle of rank 2. We showed that any such bundle on \({\mathbf{P}}^1\) is of the form \({\mathcal{O}}(a) \oplus {\mathcal{O}}(b)\), and tensoring by \({\mathcal{O}}(-a)\) yields \({\mathcal{O}}\oplus {\mathcal{O}}(n)\) where \(n\in {\mathbf{Z}}_{\geq 0}\). We write \({ \mathbf{F} }_n = {\mathbf{P}}_{{\mathbf{P}}^1}({\mathcal{O}}\oplus {\mathcal{O}}(n))\). Note that \({ \mathbf{F} }_0 = {\mathbf{P}}^1\times {\mathbf{P}}^1\). We can write \(\operatorname{Pic}({ \mathbf{F} }_0) = {\mathbf{Z}}e + {\mathbf{Z}}f\) where \(e,f\) are fibers of the two projections. These satisfy \(e^2=f^2=0\) and \(ef=1\). Any effective curve \(C \subseteq { \mathbf{F} }_0\) can be written as \(ae + bf\) for some \(a,b\geq 0\). Why this is true: note that \(C.f = a\geq 0\) because \({\left\lvert {f} \right\rvert}\) is basepoint free. Alternatively, check that \(p_*(f) = 0\) where \(p\) is the projection onto \({\mathbf{P}}^1\), and \(p_*(e) = [{\mathbf{P}}^1]\). Thus \(p_*(C) = [{\mathbf{P}}^1]\cdot\deg(C\to {\mathbf{P}}^1)\geq 0\).

Note that the classical minimal model is not unique. If \(n\neq 1\), there are no \(-1\) curves. If \(n=1\) the \(-1\) curve can be blown down to obtain \({\mathbf{P}}^2\). Thus \({ \mathbf{F} }_n\) is minimal when \(n\geq 1\), but all of the \({ \mathbf{F} }_n\) are birational.

If \(f\) contracts a curve \(E\), it must be the case that \(E^2 < 0\). Why: write \(f^* C = C" + mE\), then \(E.f^*C = f_* E.C = 0.C = 0\) on one hand, and \(E.f^*C = (C' + mE)E = C'.E + mE^2\), and since \(C'.E > 0\) and \(m > 0\), one must have \(E^2 < 0\).

Question: conversely, if \(\cup E_i \subseteq S\) and \([E_i.E_j]\) is negative definite, does there exist a contracting morphism \(S\to S'\)? Answer: yes, if \(S'\) is a complex analytic surface. This in fact holds for any Moishezon variety. By Artin, the answer is yes if the configuration is rational, which is computable. This is proved constructively.

Next time: del Pezzo surfaces.

Recall that if \(S\) is a surface then \({\mathbf{T}}_S\) is a rank 2 vector bundle with dual \(\Omega_S\). We define the canonical line bundle \(\omega_S \mathrel{\vcenter{:}}=\operatorname{det}\Omega_S = \wedge^2 \Omega_S\). This is spanned by \(dx \wedge dy\) and is thus rank 1. Since this is a line bundle, it is of the form \({\mathcal{O}}_S(K_S)\) for some divisor \(K_S\), denoted that canonical divisor. The anticanonical bundle is \(\omega_S {}^{ \vee }\), the dual of the canonical, with associated divisor \(-K_S\), the anticanonical divisor. Note that any toric boundary divisor is anticanonical. This divisor is not unique.

Recall that a line bundle \(L\) is very ample if the associated rational map \(\phi: S\to {\mathbf{P}}H^0(L) {}^{ \vee }\) (where \(x\mapsto {\mathbf{P}}(s\mapsto s(x))\)) is well-defined everywhere. Equivalently, \(L\) is basepoint free, i.e. for every \(x\in S\) there is a section \(s\) where \(s(x)\neq 0\). This follows because \([0:\cdots :0]\) is not a point in \({\mathbf{P}}^N\), and this condition guarantees that at least one coordinate is nonzero. In this case, \(\phi\) is an embedding. A divisor \(L\) is very ample iff \({\mathcal{O}}_S(L)\) is very ample. We say \(L\) is ample if \(L^{\otimes n}\) is very ample for some \(n\in {\mathbf{Z}}_{>0}\).

In higher dimensions, any \(X\) with \(-K_X\) ample is called a Fano variety. Thus a Fano surface is by definition a del Pezzo surface. This condition roughly describes having positive curvature – determining precisely when such metrics exist uses K-stability. An interesting research question: can Fano varieties be classified? Some classification results:

• \(\dim X = 1\): \(X = {\mathbf{P}}^1\) is the only Fano.
• \(\dim X = 2\): del Pezzo surfaces, which we will classify today.
• \(\dim X \geq 3\): there are finitely many Fanos up to deformation (theorem, 1960s). In \(\dim X = 3\), there are 105. In \(\dim X = 4\), the number is unknown.

Claim: \({\mathbf{P}}^2\) is a del Pezzo surface. Check that \(-K_{{\mathbf{P}}^2} = {\mathcal{O}}_{{\mathbf{P}}^2}(3)\), which is very ample. Sections intersect the zero section along a cubic, and this has 10 global sections since these biject with degree 3 homogeneous polynomials in 3 variables \(x,y,z\) on \({\mathbf{P}}^2\). Note that \({\mathbf{P}}^2\) is toric, so choosing this as a polarization yields a moment polytope with 10 integral points: a triangle with side lengths 3. \(-K_{{\mathbf{P}}^2}\) defines an embedding \({\mathbf{P}}^2\hookrightarrow{\mathbf{P}}H^0({\mathcal{O}}(3))\) which sends \([x:y:z]\) to \([x^3:,x^2 y:,\cdots,]\), all monomials of degree 3.

If \(S\) is a del Pezzo surface, then

• \(K_S^2 > 0\), and
• \(-K_S.C > 0\) for any algebraic curve \(C\) on \(S\).

Note that \((-K_S)^2 = K_S^2\). Generally, if \(L\) is ample then \(L^2 > 0\). We define the degree of a del Pezzo surface to be \(K_S^2\). Note that conversely, any surface satisfying these properties is a del Pezzo by the Nakai-Moishezon criterion. This is proved in Hartshorne chapter 5.

Some non-obvious del Pezzo surfaces:

• \(({\mathbf{P}}^2, {\mathcal{O}}(3))\)
• \(({\mathbf{P}}^1\times {\mathbf{P}}^1, {\mathcal{O}}(2,2))\).
• \((S = \operatorname{Bl}_k {\mathbf{P}}^2, {\mathcal{O}}(-K_S))\) where \(-K_S = 3H - \sum E_i\) with \(E_i^2=-1\) the exceptional curves and \(H\in \operatorname{Pic}(S)\) is the class of a general line in \({\mathbf{P}}^2\) (avoiding the blowup points).

The first two are very ample, which is easy to check. Note that \(\operatorname{Pic}(S) = H_2(S; {\mathbf{Z}})\) in the third case, and that the \(k\) points must be in general position. Different configurations of points yield deformation-equivalent surfaces. In \(\dim X = 2\), there are thus 10 possible del Pezzo surfaces up to deformation equivalence. To compute \((-K_S)^2\), one needs to know

• \(H^2 = 1\)
• \(H.E_i = 0\)
• \(E_i E_j = 0\) for \(i\neq j\)
• \(E_i^2 = -1\)

Distributing and multiplying, one obtains \(K_S^2 = 9-k\) which is positive iff \(k\leq 8\).

A counterexample where the points are not in general position: let \(p_1,p_2,p_3 \in l\) be three points contained in a line in \({\mathbf{P}}^2\). The strict transform of \(l\) is of the form \(H - E_1 - E_2 - E_3\). Intersect \begin{align*} -K_S.(H-E_1-E_2-E_3) = (3H-E_1-E_2-E_3)(H-E_1-E_2-E_3) = 3-1-1-1=0 ,\end{align*} which violates the second condition in the definition of a del Pezzo. Similarly, taking 6 points on a conic, the strict transform is \(2H-E_1-\cdots-E_6\). Intersecting as above yields \(6-1-1-1-1-1-1 = 0\). One could formulate similar conditions on higher numbers of points, but we already require \(k \leq 8\) and the next condition would be on 9 points.

Denote by \(S_d\) a del Pezzo of degree \(d\), obtained by blowing up \(k=9-d\) points in general position in \({\mathbf{P}}^2\). How to remember: \(d=9\) yields \(k=0\) points, and \(\deg {\mathcal{O}}_{{\mathbf{P}}^2}(3) = 3^2=9\). By assumption \(-K_{S_d}\) is ample, and it will be very ample if \(3\leq d\leq 9\). This corresponds to blowing up \(k\leq 6\) points. One can calculate \(h^0(-K_{S_d}) = 10-(9-d) = d+1\), so \(S_d \hookrightarrow{\mathbf{P}}^d\) where \(3\leq d\leq 9\). This is referred to as the anticanonical embedding of \(S_d\) into \({\mathbf{P}}^d\).

One could ask if this is true for higher dimensional Fano varieties. It fails in dimension 3 – not every Fano threefold is rational.

Last time: del Pezzo surfaces, where \(-K_S\) is ample. Note that for del Pezzos, \(-K_S\) is also effective, and is of the form \(S = {\mathbf{P}}^1\times {\mathbf{P}}^1\) or \(S = \operatorname{Bl}_k {\mathbf{P}}^2\) where \(0\leq k \leq 8\).

Generally, \(L\) is an effective line bundle iff \(L\) has a nonzero section \(s\). By intersecting \(s\) with the zero section of \(L\), one obtains a divisor \(D\). In the 1-dimensional case, this is the correspondence \(D\rightleftharpoons L\mathrel{\vcenter{:}}={\mathcal{O}}(D)\).

Question: is \(-K_X\) effective for any (smooth) Fano of any dimension over \({\mathbf{C}}\)?

• Dimension 2: true.
• Dimension 3: true (Sokurov, 1980).
• Dimension 4: true (Kawamata, 2000).
• Dimension \(n\geq 5\): open.

Generally ample does not imply effective. For example, let \(C\) be a genus 2 curve and \(L\) a degree 1 line bundle. Any positive degree line bundle on a curve is ample. Consider the moduli space \(\operatorname{Pic}^1(C)\) of degree 1 line bundles on \(C\). This is isomorphic to the Jacobian of \(C\), an abelian variety of dimension 2. If \(L\) is effective as a line bundle, then \(L\cong {\mathcal{O}}(D)\) with \(D\) effective. By definition, \(D\) is effective iff \(D =\sum a_i D_i\) with \(a_i > 0\). Since \(\deg L = 1\), without loss of generality we can write \(a_1 = 1\) and thus \(a_i = 0\) for \(i\neq 1\). So \(L = {\mathcal{O}}(a_1 p)\) where \(p\in C\) is a point. Thus the moduli space of such bundles is in bijection with the points of \(C\), and is dimension 1. Since \(\operatorname{Pic}^1(C)\) includes ample line bundles, there are many ample but non-effective bundles.

For any algebraic curve \(C\), there is an Abe-Jacobi map given by \(\phi: \operatorname{Sym}^d(C)\to \operatorname{Pic}^d(C)\). Note that \(\operatorname{Sym}^d(C) = C^d/S_d\) where \(S_d\) is the symmetric group, and is the moduli space of degree \(d\) effective divisors on \(C\). On the other hand, \(\operatorname{Pic}^d(C)\) is the moduli space of degree \(d\) line bundles. This map is given by \(D\mapsto {\mathcal{O}}(D)\), and is not surjective in general.

Recall that \(X\) is rational if \(X\overset{\sim}{\dashrightarrow}{\mathbf{P}}^n\). In dimension 1, \({\mathbf{P}}^1\) is the only smooth projective rational curve. In dimension 2, any del Pezzo is rational. Questions:

• Can we classify all rational surfaces?
• Can we find criteria to decide if a surface is rational or not?

Recall that \(\Omega^k_X\) is the vector bundle of algebraic \(k\)-forms on a smooth projective variety \(X\) of dimension \(n\). These are used to define Hodge numbers \(h^{k, 0}(X) = \dim H^0(\Omega^k_X)\), which are birational invariants.

For \(X = {\mathbf{P}}^n\), one has \(h^{k, 0} = 0\) for \(k\neq 0\) and \(h^{0, 0} = 1\). For \({\mathbf{P}}^2\), this yields the following Hodge diamond:

\begin{align*}\begin{tikzcd} && 1 \\ & 0 && 0 \\ 0 && 1 && 0 \\ & 0 && 0 \\ && 1 \end{tikzcd}\end{align*} > Link to Diagram

Thus any rational surface should have a Hodge diamond of the following form:

\begin{align*}\begin{tikzcd} && {h^{2,2}} \\ & {h^{2,1}} && {h^{1,2}} \\ 0 && {h^{1,1}} && {h^{2,0}} \\ & 0 && {h^{1,0}} \\ && 1 \end{tikzcd}\end{align*} > Link to Diagram

One can conclude, for example, that K3 surfaces are not rational.

Question: is any surface \(S\) with \(h^{1, 0}(S) = h^{2,0}(S) = 0\) necessarily rational? Answer: no, consider Enriques surfaces, which are not rational:

\begin{align*}\begin{tikzcd} && 1 \\ & 0 && 0 \\ 0 && 10 && 0 \\ & 0 && 0 \\ && 1 \end{tikzcd}\end{align*} > Link to Diagram

We instead introduce more refined birational invariants in terms of \(h^0( (\Omega^k)^{\otimes m} )\) for \(m > 0\). For \({\mathbf{P}}^n\), these vanish for all \(m\). For \(\dim X = k\), we define the plurigenus \begin{align*} P_m(X) \mathrel{\vcenter{:}}= H^0( (\Omega^k_X)^{\otimes m}) .\end{align*} For \(m=1\), this is referred to as the geometric genus.

Let \(S\) be a smooth projective surface. Then \(S\) is rational iff

• \(q \mathrel{\vcenter{:}}= h^{1, 0} = 0\) (vanishing irregularity)
• \(P_2 = h^0(\omega_S^{\otimes 2}) = 0\).

In the forward direction, this is clear since these are birational invariants and \(q({\mathbf{P}}^2) = h^{1, 0}({\mathbf{P}}^2) = 0\) and \(P_2({\mathbf{P}}^2) = 0\). The other direction requires a difficult proof. We’ll assume \(S\) is minimal, i.e. \(S\) does not contain a smooth rational curve \(C\) with \(C^2 = -1\) (which can be contracted).

General philosophy: \(C\) is given by \(V(f)\) for some polynomial \(f\), and since \(C^2 \geq 0\) it can be deformed to \(V(f')\). One can form the pencil \(\left\{{uf + vf'}\right\}\).

Consider \(S = {\mathbf{P}}^2\), take \(C\) to be any line in \({\mathbf{P}}^2\). Then \({\mathcal{O}}(-3).[C] = -3 < 0\), and \(L\mathrel{\vcenter{:}}={\mathcal{O}}(K_S + C) \cong {\mathcal{O}}(-2)\) has no sections.

Why the lemma implies the proposition: let \(C\) be a curve satisfying the conditions of the lemma, then we claim \(C\cong {\mathbf{P}}^1\). The proof uses adjunction and Riemann-Roch: \(2g-2 = C(C+K_S)\) where \(g\) is the arithmetic genus, and \(\chi({\mathcal{O}}(D)) = {1 \over 2}D(D-K_S) + \chi({\mathcal{O}})\). Note that \(\chi({\mathcal{O}}) = 1 - h^{1, 0} + h^{2, 0} = 1-0+0\) by assumption, so \(\chi({\mathcal{O}}(D)) = {1\over 2}D(D-K_S) + 1\). Take \(D \mathrel{\vcenter{:}}= K_S + C\). Recall that by Serre duality, \(h^2({\mathcal{O}}(D)) = h^0({\mathcal{O}}(K_S-D))\). Thus \(h^2({\mathcal{O}}(K_S + C)) = h^0({\mathcal{O}}(-C)) = 0\) since \(-C\) is not effective. As a result, \(\chi(K_S + C) = h^0(K_S + C)-h^1(K_S + C)\). We know by assumption that \(h^0(K_S + C)\), so \(\chi(K_S +C)\leq 0\). Substituting this into Riemann-Roch yields \({1\over 2}(K_S + C)C + 1 \leq 0\). Solving for \(g\) in the adjunction formula yields \(g = {1\over 2}(K_S + C)C + 1\), so \(g\leq 0\). The arithmetic genus can not be negative for an irreducible curve, hence \(g=0\) and \(C\cong {\mathbf{P}}^1\).

It remains to show that \(C^2 \geq 0\). By adjunction, \(-2 = 2g-2 = C(K_S + C)\) since \(g=0\). We assumed \(C.K_S < 0\), so \(C^2 + C.K_S < C^2\), so \(C^2 = -1\) or \(C^2 \geq 0\). The former case is ruled out because we assumed \(S\) to be minimal.

On the proof of the lemma: it is broken into the following three cases,

• \(K^2 < 0\),
• \(K^2 = 0\),
• \(K^2 > 0\).

We’ll consider the case \(K^2 = 0\). We first claim \(-K_S\) is effective, i.e. \(h^0(-K_S) \neq 0\). Applying Riemann-Roch to \(-K_S\) yields \(\chi(-K_S) = {1\over 2}(-K)(-K-K) + 1 = -K^2 + 1= 1\) since \(K^2 = 0\). By Serre duality, \(h^2(-K_S)= h^0(K_S - (-K_S)) = h^0(2K_S)\), but note \({\mathcal{O}}(2K_S) = {\mathcal{O}}(K_S)\otimes{\mathcal{O}}(K_S) = \omega_S^{\otimes 2}\), so \(h^0(2K_S) = P_2 = 0\) by assumption. Thus \(h^0(-K_S) = 1 + h^1(-K_S) > 0\).

Claim: let \(H\) be a very ample divisor, which is in particular effective. Then \(-K_S.H > 0\) since \(-K_S\) is effective, so \(K_S.H < 0\). We’ll show \(H + nK_S\) is not effective if \(n\) is large enough. Consider \(H(H + nK_S) = H^2 + nHK_S\); sine \(HK_S < 0\), this is negative for some \(n\) since \(H^2\) is a fixed number. If \(H + nK_S\) were effective, this would have to be positive. So there exists a unique minimal \(n_0\) such that \(H + n_0 K_S\) is not effective for any \(n\geq n_0\). So take \(C \mathrel{\vcenter{:}}= H + n_0 K_S\). One then checks \(K_S.C = K_S.H < 0\) since \(H^2 = 0\) and \(-K_S\) is effective with \(H\) ample. Furthermore \(K_S + C= K_S + H + n_0 K_S = H + (n_0 + 1)K_S\) which is not effective by definition of \(n_0\), thus \({\left\lvert {K_S + C} \right\rvert} = \emptyset\).

Note that we didn’t show \(C\) was irreducible. If it is not, then take any irreducible component \(C'\) with \(C'.K_S < 0\), which must exist since \(K_S.C < 0\).

Recall that a minimal surface is a smooth projective surface which does not contain any smooth rational \((-1)\)-curves. Castelnuovo’s contraction theorem implies that any smooth projective surface is birational to a minimal surface. Goal: classify all smooth projective surfaces up to birational equivalence. A related question: in any birational equivalence class, is there a unique minimal surface? Or can two minimal surfaces be birational? Answer:

• No for rational surfaces,
• Yes for non-ruled surfaces.

Recall that a ruled surface is birational to \(C\times {\mathbf{P}}^1\) for \(C\) a smooth projective curve. Note that rational implies ruled, since \({\mathbf{P}}^2 \overset{\sim}{\dashrightarrow}{\mathbf{P}}^1\times {\mathbf{P}}^1\) which is ruled. Every non-ruled surface admits a unique minimal model up to isomorphism.

Let \(S, S'\) be non-ruled surfaces and \(\phi: S'\overset{\sim}{\dashrightarrow}S\) be a birational map. We want to show that \(S' \cong S\). Note that \(\phi\) is a composition of blowups and blowdowns, so there exists some \(\widehat{S}\) with \(f: \widehat{S}\to S\) and \({\varepsilon}: \widehat{S}\to S'\) where \(f, {\varepsilon}\) are compositions of blowups. Choose a diagram such that \({\varepsilon}\) is composed of a minimal number \(n\in {\mathbf{Z}}_{\geq 0}\) of blowups. By cases: if \(n=0\), then \(S'\cong \widehat{S}\), and \(f:S'\to S\) is a composition of blowups. The last blowup would introduce an exceptional curve, contradicting the fact that \(S'\) is minimal. Thus \(f\) is an isomorphism.

Suppose now that \(n\neq 0\); we’ll show that this can never happen. Let \(E\) be the exceptional curve of the blowup \({\varepsilon}\) in \(\widehat{S}\). Consider \(f(E)\), then either

• \(f(E)\) is contracted, so \(f(E)\) is a point, or
• \(f(E)\) is a curve.

The first case contradicts the minimality of \(n\) by removing the blowup/blowdown of \(E\) from the diagram. In the second case, we’ll show such a curve can not exist, reaching a contradiction.

We now apply this lemma to the previous proof. Recall that \({\varepsilon}: \widehat{S}\to S\) produces an exceptional curve \(E = \widehat{\Gamma}\) where the last blowup \({\varepsilon}_n\) produces \(E\). Let \(C = f(E)\). Apply the lemma to each blowup \(f_1,\cdots, f_m\) by setting \(\widehat{\Gamma }= E\). By the adjunction formula, \(2g(E)-2 = E.(E+K_{\widehat{S}})\), and since \(g(E) = 0\) one obtains \(E.K_{\widehat{S}} = -1\). Thus \(C.K_S \leq -1\) by the lemma. Claim: \(C.K_S \neq -1\), so the inequality is strict. If \(K_S.C = -1\), then \(K_{\widehat{S}}.E = K_S.C\) and this only happens if \(E\) does not intersect the other exceptional divisor \(E\). So \(f(E) \cong C \cong E\), so \(C\cong {\mathbf{P}}^1\) with \(C^2 = -1\), but this contradicts minimality of \(S\). Thus \(C.K_S \leq -2\).

By adjunction, \begin{align*} 2g(C)-2 = C.(C+K_S) = C^2 + C.K_S \implies C^2 = 2g-2-C.K_S \implies C^2\geq 0 \end{align*} since \(C.K_S \leq -2\).

Hence \(P_m(S) = 0\), and in particular \(P_2(S) = 0\). Cases:

• \(h^{1, 0}(S) = 0\), which implies \(S\) is rational by Castelnuovo, hence ruled, a contradiction.
• \(h^{1, 0}(S) \neq 0\), which we claim also can not happen.

Assume the second case. Black box: for any surface \(S\), there is a natural map \(\psi: S\to \operatorname{Alb}(S)\) where \(\psi(S)\) is a curve of genus \(h^{1, 0}\). Moreover, \(\psi\) is generically finite, and \(\operatorname{Alb}(S)\) is generally a genus \(g\) abelian variety. Since \(C\) is rational, it must be a fiber of \(\psi\). One can show \(S\cong \operatorname{Alb}(S) \times {\mathbf{P}}^1\), showing \(S\) is ruled, a contradiction. This forces \(n=0\) and \(S\cong S'\).

Next time: the Albanese variety.

Let \(S, S'\) be non-ruled surfaces, then any birational map \(S\overset{\sim}{\dashrightarrow}S'\) is an isomorphism. Factoring as a composition of \(n\) blowups and \(m\) blowdowns, we saw that if \(n=0\) this is true. For \(n\geq 1\), we showed existence of a curve \(C\) with \(K_S.C \leq -2\) and \(C^2 \geq 0\), which implied \(P_m(S) = 0\), and in particular \(P_2(S) = 0\). There were two cases:

• \(h^{1, 0}(S) = 0 \implies S\) is rational, hence ruled, a contradiction, or
• \(h^{1, 0}\neq 0\).

In the latter case, we have a natural map \(\psi: S\to \operatorname{Alb}(S)\), which is an abelian variety. The image \(\psi(S)\) is a genus \(g\) curve where \(g = h^{1, 0}(S)\). Note that a priori, one could have \(\dim \psi(S) = 0,1,2\), so we argue that \(\dim \psi(S)\neq 0, 2\). If \(\psi(S)\) is a point, the universal property of the Albanese yields a contradiction. If \(\psi(S)\) is a surface, then \(\psi\) is generically finite and \(P_2(S) = P_2(\operatorname{Alb}(S))\neq 0\).

Let \(X\) be any smooth projective variety, then there is an abelian variety \(\operatorname{Alb}(X)\). This can be written as \(H^0(\Omega_X) {}^{ \vee }/\iota(H_1(X; {\mathbf{Z}}))\) where \(\iota: H_1(X;{\mathbf{Z}})\to H^0(\Omega_X) {}^{ \vee }\) is defined by \(\gamma \mapsto (\omega\mapsto \int_\gamma \omega)\). It is a theorem that the image is always a lattice \(\Gamma\), so \(\Gamma \otimes{\mathbf{C}}= H^0(\Omega_X) {}^{ \vee }\). The proof uses deep facts from Hodge theory.

Recall that \(h^{1, 0} = \dim_{\mathbf{C}}H^0(\Omega_X) \mathrel{\vcenter{:}}= n\), so \(H^0(\Omega_X)\cong {\mathbf{C}}^n \cong {\mathbf{R}}^{2n}\). Writing \(H_1(X;{\mathbf{Z}}) = {\mathbf{Z}}^{b_1} + T\) where \(T\) is torsion, note that \(\iota(T) = 0\). Note also that \(b_1 = h^{1, 0} + h^{0, 1}\). Moreover \(\iota({\mathbf{Z}}^{b_1})\cong {\mathbf{Z}}^{b_1}\). Thus \(\operatorname{Alb}(X) \cong {\mathbf{R}}^{2n}/{\mathbf{Z}}^{2n}\cong (S^1)^{2n}\). There is a natural complex structure on \(\operatorname{Alb}(X)\), and moreover is smooth and projective. The proof again uses Hodge theory. By GAGA, \(\operatorname{Alb}(X)\) is the unique variety with this underlying complex manifold structure.

Recall \(\operatorname{Pic}^0(X)\) is the connected component of \(\operatorname{Pic}(X)\) containing the trivial line bundle. A theorem of Grothendieck shows that \(\operatorname{Pic}^0(X)\) is an abelian variety of dimension \(h^{1, 0}\). We also have \(\operatorname{Alb}(X) = \operatorname{Pic}^0(X) {}^{ \vee }\) – more generally, for any abelian variety \(A\), one defines \(A {}^{ \vee }\mathrel{\vcenter{:}}=\operatorname{Pic}^0(A)\) since \(\operatorname{Pic}^0(\operatorname{Pic}^0(A)) = A\). We thus regard \(\operatorname{Alb}(X)\) as the “smallest” abelian variety associated to \(X\). The map \(\psi: X\to \operatorname{Alb}(X)\) can be given by \(x\mapsto (\omega\mapsto \int_x^p \omega)\) where \(p\) is a fixed choice of point on \(X\). The universal property is that for any abelian variety \(A\), any map \(X\to A\) factors as \(X\to \operatorname{Alb}(X)\to A\).

Recall Castelnuovo’s rationality criterion: let \(S\) be a smooth projective surface, then \(S\) is rational iff \(q\mathrel{\vcenter{:}}= h^1({\mathcal{O}}_X) = h^0(\Omega_X) = 0\) and \(P_2 \mathrel{\vcenter{:}}= h^0(2K_S) = 0\). As a corollary, \(S\) is rational iff \(S\) is unirational, i.e. there is a dominant map \({\mathbf{P}}^2 \dashrightarrow S\) of some degree \(d\). We discussed minimal surfaces, for which there are two cases:

• \(S\) is ruled, so \(S \overset{\sim}{\dashrightarrow}C\times {\mathbf{P}}^1\) for some curve \(C\). This implies that a generic point is contained in a rational curve.
• \(S\) is not ruled. In this case, there are few rational curves.

We showed that if \(S\) is not ruled, there exists a unique minimal model in every birational equivalence class. That is, if \(f: S'\overset{\sim}{\dashrightarrow}S\) with \(S, S'\) minimal surfaces, then \(f\) is an isomorphism. If \(S\) is ruled, minimal models are not unique – blow up any point in a fiber of \(S\to C\) to transform one \((0)\)-curve to a union of two \((-1)\)-curves, and blowdown the other curve to get another ruled surface.

Important trick: suppose \(C\) is an effective irreducible curve where \({\left\lvert {K_S + C} \right\rvert} = \emptyset\). Suppose also that \(\chi({\mathcal{O}}_S) = 1\), noting that \(\chi({\mathcal{O}}_S) = 1-q+p_2\) which vanishes for rational surfaces. Then \(C\cong {\mathbf{P}}^1\).

Recall that \(\chi({\mathcal{O}}(D)) = \chi({\mathcal{O}}_S) + {1\over 2}D(D-K_S)\). Applying this to \(D = K+C\) yields \(\chi(K+C) = 1 + {1\over 2}(K+C)C = p_a(C)\). On the other hand, \(\chi(K+C) = h^0(K+C) - h^1(K+C) +h^0(-C)\). Here \(h^0(K+C) = h^0(-C) = 0\), so \(\chi(K+C)\leq 0\), and thus \(p_a(C)\leq 0\), forcing \(p_a(C) = 0\) and \(C\cong {\mathbf{P}}^1\).

Recall that if \(\nu: \tilde C\to C\) is the normalization, then \(p_a(C) \geq p_a(\tilde C) = g(\tilde C)\).

Consider \(S = {\mathbf{P}}^2\); the key is to look at minimal rational curves on \(S\). Note that this generalizes to higher dimensions, viz. Mori theory. For \({\mathbf{P}}^2\), these minimal curves will be lines. If \(C\) is a line, then \(KC = -3 < 0\) and \({\left\lvert {K+C} \right\rvert} = {\left\lvert {-3H + H} \right\rvert} = \emptyset\).

For \(S = { \mathbf{F} }_n\), note that this is a \({\mathbf{P}}^1\) fibration over \({\mathbf{P}}^1\) and the minimal rational curves are its fibers. Letting \(C\) be a fiber, we have \(C^2 = 0\) and by the genus formula, \(KC = -2\). Moreover \({\left\lvert {K+C} \right\rvert} = \emptyset\), using the following observation:

Suppose \(A\) is an irreducible effective curve with \(A^2 \geq 0\) with \(D.A < 0\). Then \(D\) is not effective and \({\left\lvert {D} \right\rvert} = \emptyset\). This follows from \(D = nA + \sum a_i B_i\) with \(AB_i \geq 0\) and \(nA^2\geq 0\).

One can then check that if \(K+C\) is effective, then its restriction to a fiber is still effective. However, \(K_{{\mathbf{P}}^1} = {\mathcal{O}}_{{\mathbf{P}}^1}(-2)\) is not effective.

We now prove that if \(S\) is rational and minimal, then \(S\cong {\mathbf{P}}^2, { \mathbf{F} }_n\). Consider the set of irreducible curves \(\left\{{C \cong {\mathbf{P}}^1 \subseteq S {~\mathrel{\Big\vert}~}C^2 \geq 0}\right\}\neq \emptyset\). This is nonempty by a previous result on existence of special curves on such surfaces. Consider the subset of such curves with minimal square \(C^2 = m\). Fix a hyperplane section \(H\), so \(C.H = \deg(C)\), and consider the further subset with minimal \(C.H\).

Next time: finish step 1, understand the minimal surfaces which are rational and not ruled.

For minimal surfaces, there is a division:

• Non-ruled surfaces: \(\exists !\) minimal model in every birational class. We showed there is a sequence of contractions \(S\to\to\to S_{\min}\) where \(S_{\min}\) has no \((-1)\)-curves. Therefore our next step will be classifying such minimal surfaces.

• Ruled surfaces: minimal models are never unique. There are two cases:

  • \(q\mathrel{\vcenter{:}}= h^1({\mathcal{O}}_S) = 0\): regular surfaces. Minimal surfaces are \({\mathbf{P}}^2\) and \({ \mathbf{F} }_n\) for \(n\neq 1\), thus rational.
  • \(q\neq 0\): non-regular surfaces. Minimal surfaces are \({\mathbf{P}}_C({\mathcal{E}})\) with \(\operatorname{rank}{\mathcal{E}}= 2\) and \({\mathcal{E}}\) a vector bundle, i.e. geometrically ruled surfaces (every fiber is \({\mathbf{P}}^1\)). Here \(g(C) = q\) and \(p: S\to C\) is the Albanese map.

Last time: suppose \(S\) is a rational surface, we considered minimal rational curves \(C\) on \(S\), i.e.  \begin{align*} \left\{{C\cong {\mathbf{P}}^1 \subseteq S {~\mathrel{\Big\vert}~}C^2 \geq 0}\right\} \supseteq\left\{{C{~\mathrel{\Big\vert}~}C^2 = m \text{ is minimal }}\right\} \supseteq\left\{{C {~\mathrel{\Big\vert}~}\deg(C) \mathrel{\vcenter{:}}= C.H \text{ is minimal} }\right\} .\end{align*} Then every \(D\in {\left\lvert {C} \right\rvert}\) is smooth, irreducible, and \(D\cong {\mathbf{P}}^1\), i.e. curves in \({\left\lvert {C} \right\rvert}\) do not split. For example, conics in \({\mathbf{P}}^2\) degenerate into unions of lines and thus do split.

This argument generalizes to higher dimensions. Note that \({\left\lvert {C} \right\rvert}\) is a covering family of \(S\) of rational curves with minimal degree. See e.g. Mori’s Annals paper proving Hartshorne’s conjecture, which started the Mori program, or papers that use the bend-and-break technique. E.g. if \(X\) is a smooth projective Fano variety, so \(-K_X\) is ample, then \(X\) is covered by rational curves and such covering families exist.

Recall that ruled non-regular surfaces (\(q\neq 0\)) are birational to \(C\times {\mathbf{P}}^1\) where \(g(C) = q\). First factor \(S\to C\times {\mathbf{P}}^1\) as \(S\to\to\to S'\to\to\to C\times {\mathbf{P}}^1\) as a composition of blowdowns and blowups. Note that blowing up points in fibers of \(S\to C\times {\mathbf{P}}^1\) yields unions of \({\mathbf{P}}^1\)s. The claim that \(E_i\) must be contained in a fiber, and thus not horizontal. This is because \(g(E_i) = 0\) but \(g(C) = q > 0\), and \({\mathbf{P}}^1\) can not map to a curve of higher genus. One can show this using Riemann-Hurwitz, since \(\deg K_C = 2g(C) - 2\) and \(\deg K_{{\mathbf{P}}^1} = -2 < 0\). Alternatively, \(f^* \Omega_C \subset \Omega_E\) which, by taking dimensions, yields \(g(C) \leq g(E)\). Thus the exceptional curves are vertical. Since there is a fibration \(S'\to C\), there are thus fibrations to \(C\) after each blowdown and \(S\to C\) fibers over \(C\).

Note that the \(E_i\) are not equivalent to fibers, because the intersection form of the exceptional curves is negative semi-definite. Next chapter: ruled, non-regular surfaces, then we move on to non-ruled surfaces.

Recall \(p_g \mathrel{\vcenter{:}}= h^0(K_S)\), so if \(p_g = 0\) then there is not effective \(D\sim K_S\). For ruled surfaces, all \(P_m \mathrel{\vcenter{:}}= h^0(mK_S) = 0\), so no multiple of \(K_S\) is effective. Use the useful observation: if \(C^2 \geq 0\) with \(mK.C < 0\) then \(mK\) is not effective. Take \(C\) to be a fiber \(F\), then \(mK_S.F + F^2 = 2g(F) - 2\) and so \(mK_S.F = -2 < 0\).

Noether’s formula: \(\chi({\mathcal{O}}_S) = {K^2 + \chi(S)\over 12} = {c_1^2 + c_2\over 12}\), which was derived from Hirzebruch-Riemann-Roch, \begin{align*} \chi(D) = \int_X e^D \operatorname{Td}({\mathbf{T}}_X) ,\end{align*} where you multiply the corresponding power series and take the top degree terms. We have \(\chi({\mathcal{O}}_S) = 1-q+p_g\) in general, and note that \(p_g=0\) in this case. Note that \(\chi(S) = \sum (-1)^i b_i = 2-2b_1 + b_2\) by Poincare duality, and \(b_1 = h^{1, 0} + h^{0, 1} = 2q\) by the Hodge decomposition. Substituting these facts into Noether’s formula yields \begin{align*} 12(1-q) = K^2 -4q + b_2 \implies K^2 = 10-8q - b_2 .\end{align*}

Surfaces with \(p_g=0\) and \(q\geq 1\), i.e. ruled irregular surfaces (plus a little more). The next main goal: minimal non-ruled surfaces.

Examples where \(K\) is not effective but \(2K\) is effective: Enriques surfaces.

Proving the proposition, the main tool: the Albanese map \(p: S\to \operatorname{Alb}(S)\) which is an abelian variety of dimension \(q\). Note that \(p(S)\) generates \(\operatorname{Alb}(S)\), and \(p\) has connected fibers. If \(q\geq 1\) then \(\dim p(S) = 1,2\). If \(p_g = 0\), then \(\dim p(S) = 1\). Any abelian variety is of the form \(A = {\mathbf{C}}^g/{\mathbf{Z}}^{2g}\), and the differential forms are those with constant coefficients, e.g. \(dz_1,\cdots,dz_n\), \(dz_1\wedge dz_2 \cdots\), etc. These are translation invariant and thus descend to the quotient. One then produces a differential form in \(\operatorname{Alb}(S)\) which pulls back to \(S\), yielding a nontrivial section of \(\Omega^2(S)\), forcing \(\dim p(S) = 1\). In this case, \(p: S\to B\) where \(B\) is a smooth curve of genus \(q\). Note that if a single fiber is \({\mathbf{P}}^1\) then \(S\) is ruled. We’re trying to rule out fibers with higher genus.

What remains to do: look at the “a little more” case. Two cases:

• The genus of the general fiber satisfies \(g(F)\geq 2\), then \(p\) is a smooth map yielding a family of elliptic curves over an elliptic curve
• \(g(F) = 1\), there may be multiple fibers. These appear in the classification of minimal models with \(\kappa = 0\). Proofs here involve the topological Euler characteristic.

Recall that we are considering minimal surfaces \(S\) with \(q\geq 1, p_g = 0\). We had \(K^2 = 10-8q-b_2\), and concluded that if \(K^2 < 0\) then \(S\) is ruled. We consider now the extreme case where \(K^2 = 0\), so \(q=1, p_g = 0, b_2 = 2\), and \(S\) is not ruled. Consider the Albanese morphism \(p: S \to B \subseteq \operatorname{Alb}(S)\) with \(g(B) = q\); we’d like to show every fiber is irreducible. If any fiber is reducible, this forces \(b_2\geq 3\) by taking two components along with a horizontal part. If any fiber is \({\mathbf{P}}^1\), then \(S\) is ruled and birational to \(B\times {\mathbf{P}}^1\). Since \(S\) is reduced and \(p\) is nonconstant onto a smooth curve, \(p\) is flat.

In the \(g=1\) case, the fibration \(p\) is almost smooth. Note that isotrivial families aren’t necessarily cartesian products. Moreover, we claim that \(p\) is never the trivial family \(S = B\times F\). If it were, \(K_S = \pi_1^* K_B + \pi_2^* K_F\) is effective, contradicting \(p_g = 0\).

To study this, we consider the topological Euler characteristic, defined as \(\chi_{{\mathsf{Top}}}(X) = \sum_{i=0}^{\dim_{\mathbf{R}}X} (-1)^i \dim H^i_c(X)\) for \(X\) any algebraic variety over \({\mathbf{C}}\), not necessarily compact. Fact: if \(X\supseteq Z\) a closed set with open complement \(U \mathrel{\vcenter{:}}= X\setminus Z\), one has \(\chi_{\mathsf{Top}}(X) = \chi_{\mathsf{Top}}(Z) + \chi_{\mathsf{Top}}(U)\). As a corollary, for a fibration like \(p\), one has \begin{align*} \chi_{\mathsf{Top}}(S) = \chi_{\mathsf{Top}}(B)\cdot \chi_{\mathsf{Top}}(F_{\mathrm{gen}}) + \sum_{\mathrm{finite}}\chi_{\mathsf{Top}}(F_s)- \chi_{\mathsf{Top}}(F_{\mathrm{gen}}) \end{align*} where \(F_s\) is the special fiber. Note that there is a LES \(H^i_c(U)\to H^i_c(X) \to H^i_c(Z)\) given by including cycles along \(Z\hookrightarrow X\) and restricting cycles to the open subset \(U \subseteq X\).

Note that \(b_2 = 2q\) by Hodge conjugate symmetry, and so in our case \begin{align*} \chi_{\mathsf{Top}}(S) = 1-b_1+b_2-b_3+1 = 2 - 2b_1 + b_3 = 4-4q=0 .\end{align*} If \(C\) is a smooth curve of genus \(g\), \begin{align*} \chi_{\mathsf{Top}}(C) = 1-b_1+1 = 2-2g .\end{align*} Note that when \(C\) is smooth, \(\chi({\mathcal{O}}_C) = h^0({\mathcal{O}}_C)-h^1({\mathcal{O}}_C) = 1-g\), and so \(\chi_{\mathsf{Top}}(C) = 2\chi({\mathcal{O}}_C)\).

Claim: \(\chi_{\mathsf{Top}}(F_s) > \chi_{\mathsf{Top}}(F_{\mathrm{gen}})\), so the finite sum in \(\chi_{\mathsf{Top}}(S)\) is nonzero. Equality only holds in the \(g=1\) case of the above theorem. This will prove the first part of the theorem. Let \(F\) be a fiber, then there is a SES \({\mathcal{O}}_S(-F) \hookrightarrow{\mathcal{O}}_S\twoheadrightarrow{\mathcal{O}}_F\) given by restriction to a fiber. Thus \[\begin{align*} \chi({\mathcal{O}}_F) &= \chi({\mathcal{O}}_S) - \chi({\mathcal{O}}_S(-F)) &= \chi({\mathcal{O}}_S) - \qty{\chi({\mathcal{O}}_S) + {F(F+K)\over 2}} &= {-FK\over 2} .\end{align*}\]

By the same argument, \(\chi({\mathcal{O}}_C) = {-CK\over 2} = {-FK\over 2n}\) since \(C^2 = 0\). Consider the case \(g(F_{\mathrm{gen}}) \geq 2\), then \(\chi({\mathcal{O}}_C)-\chi({\mathcal{O}}_{F_{\mathrm{gen}}}) > 0\). Similarly when \(g=1\). It remains to show smoothness.

Fact: any proper non-isotrivial family of curves of genus \(g\geq 1\) must degenerate, i.e. there must be a singular fiber. A similar statement holds for abelian varieties. For \(g\geq 2\), consider \({\mathcal{A}_g}\) as a coarse space. Adding a level structure for \(m\geq 3\) yields a fine space \({\mathcal{A}_g}(m)\), where a level structure is an isomorphism \(A[m] { \, \xrightarrow{\sim}\, }({\mathbf{Z}}/m{\mathbf{Z}})^{2g}\). This kills automorphisms. Note that for \(g=1\), this is easier since \(M_1 = {\mathbf{A}}^1\) and any map from a projective to an affine variety is constant.

Recall: let \(S\) be a minimal non-ruled surface with \(p_g=0\) and \(q\geq 1\), which implies \(K^2 = 0\) and \(b_2 = 2\). Then \(\pi: S\to B\) is isotrivial, where \(B\) is a curve with \(g(B) = q\). Either

• All fibers are smooth with multiplicity 1 and \(g(B) \geq 2\), or
• \(g(B) = 1\) and all fibers are either smooth or a multiple of a smooth curve.

Thus there is an etale base change such that \(S\) is a Cartesian product and the group action is free.

In the first case, \(\kappa(S) = 0\), and in the second, \(\kappa(S) \geq 1\). Note that if \(G\curvearrowright B'\) by translations and \(B'\) is elliptic, then \(B = B'/G\) is elliptic as well.

Other examples: let \(G\curvearrowright E_1\times E_2\) by a finite group action where \(G\curvearrowright E_1\) by translation and \(E_2/G \cong {\mathbf{P}}^1\). These are referred to as bielliptic surfaces. Consider \(\mathop{\mathrm{Aut}}(E)\) for \(E\) an elliptic curve. This always contain \(\mathop{\mathrm{Aut}}^0(E)\) generated by translations \(x\mapsto x+a\) for \(a\in E\). One can write \(\mathop{\mathrm{Aut}}(E) = H\rtimes\mathop{\mathrm{Aut}}^0(E)\) where \(H\) is a finite group isomorphic to \(\mathop{\mathrm{Aut}}(E, 0)\), the automorphisms that fix the origin. Generically \(H = C_2\) generated by the elliptic involution. Let \(E_\tau \mathrel{\vcenter{:}}={{\mathbf{C}}\over {\mathbf{Z}}\oplus {\mathbf{Z}}\tau}\), then \begin{align*} H = \begin{cases} C_2 & y\mapsto -y \\ C_4 & y\mapsto iy (E_i) \\ C_6 & y \mapsto \zeta_3 y (E_{\zeta_3}) \end{cases} .\end{align*}

There is a finite list of possible actions \(G\curvearrowright E_1\times E_2\) for bielliptic surfaces:

1. \(C_2\)
2. \(C_2\times C_2\)
3. \(C_4\)
4. \(C_4\times C_2\)
5. \(C_3\)
6. \(C_3\times C_3\)
7. \(C_6\)

We have

• \(2K_S \sim 0\) in cases 1,2,
• \(4K_S\sim 0\) in 3,4,
• \(3K_S \sim 0\) in 5,6,
• \(6K_S \sim 0\) in 7.

Note that as a corollary, we always have \(12K_S \sim 0\) in this case.

Claim: there is a base change of \(S\to B\) to \(S' \to B'\) such that \(S'\) is a Cartesian product. This is case 1 from earlier. An idea of the proof: there is a constant map \(B\to {\mathcal{M}_g}\to {\mathcal{A}_g}\) at the level of coarse spaces. Pass to fine spaces \({\mathcal{M}_g}(N) \to {\mathcal{M}_g}\), and \({\mathcal{A}_g}(N)\to {\mathcal{A}_g}\), which are quotients by finite groups. The latter is comprised of pairs \([C, \operatorname{Jac}(C)_N { \, \xrightarrow{\sim}\, }C_N^{2g}]\). One then considers trivializing the local system \(\R^1\pi_* C_N\).

The \(\kappa(S) = 0\) case is particularly important case for us.

Goal: classify surfaces with \(\kappa(S) = 0\). Let \(S\) be a minimal non-ruled surface, so \(P_m = 0,1\) for all \(m\) and \(P_m = 1\) for at least one \(m\). We know \(K_S\) is nef and thus \(K_S^2 = 0\).

Note that if \(X\) is a projective variety of dimension \(n\) and \(D\) is a nef divisor, then \(D^n \geq 0\). If \(A\) is an ample divisor, then Kleiman’s criterion \(D + {\varepsilon}A\) is ample \(\forall {\varepsilon}> 0\). Ampleness implies \((D + {\varepsilon}A)^n > 0\) since this is the degree of \(X\). Then take \({\varepsilon}\to 0\). Note that torsion in \(\operatorname{Pic}(X)\) is numerically zero.

Kleiman’s criterion: consider \(N_1(X) = \left\{{\sum n_i C_i}\right\}/\sim\), sums of curves modulo numerical equivalence. Similarly \(N^1(X) = \left\{{\sum n_j D_j}\right\}/\sim\), sums of divisors modulo numerical equivalence. There is an intersection product \(N_1(X)\times N^1(X) \to {\mathbf{Z}}\), and numerical equivalence means they intersect the same way. This is a perfect pairing, and \(\operatorname{rank}_{\mathbf{R}}N_1(X)\otimes{\mathbf{R}}= \operatorname{rank}{\operatorname{NS}}(X)\), so \(N^1(X) = {\operatorname{NS}}(X)/{\operatorname{tors}}\). There is a cone \({\operatorname{NE}}_1(X) \subseteq N_1(X)\otimes{\mathbf{R}}\), the Mori-Kleiman cone (or Mori cone). Any divisor \(E\) gives a linear function \(f_E\) on \({\operatorname{NE}}_1(X)\). The Kleiman criterion states that a divisor \(E\) is ample if \(f_E > 0\) on \(\overline{{\operatorname{NE}}(X)}\setminus\left\{{0}\right\}\). Thus nef divisors are limits of ample divisors.

Mori proved that if one partitions \({\operatorname{NE}}_1(X)\) by the hyperplane \(K_X\), the negative part is polyhedral, generated by extremal rays. For Fano varieties, the entire cone is polyhedral, while for abelian surfaces it is the round cone. More cones of K3 surfaces are partly polyhedral, partly round.

Note \(\chi({\mathcal{O}}_S) = 1-q +p_g\), and by Noether’s formula, \(\chi({\mathcal{O}}_S) = {c_1^2 + c_2\over 12} = {K^2 + \chi(S)\over 12}\). If \(K^2=0\), then \(12 \chi({\mathcal{O}}_S) = \chi(S) = \sum b_i = 1-2q+b_2-2q+1 = 2-4q+b_2\). Note \(b_2 = h^{0, 2} + h^{1,1}+h^{2, 0} = 2p_g + h^{1,1}\), thus \(b_2 \geq 2p_g\) and \begin{align*} 12\chi({\mathcal{O}}_S) = 2-4q+b_2 \geq 2-4q + 2p_g .\end{align*} There is a problem if \(q\) is large. Note that \(-4\chi({\mathcal{O}}_S) = -4+4q-4p_g\), and adding this to the above equation yields \begin{align*} 8\chi({\mathcal{O}}_S) = -2-4p_g + b_2 \geq -2 - 2p_g .\end{align*} Note that \(-4p_g = 0, -4\), but if \(-4p_g=-4\) then \(\chi({\mathcal{O}}_S) < 0\) would contradict this inequality.

Thus the possibilities for \(\chi({\mathcal{O}}_S) = 1-q+p_g\) are

• \(p_g = 0 \implies \chi({\mathcal{O}}_S) = 1-q\geq 0\),
• \(p_g = 1 \implies \chi({\mathcal{O}}_S) = 2-q\geq 0\).

These yield the four possibilities in the theorem.

Consider \({\mathcal{S}}\mathrel{\vcenter{:}}=\left\{{m {~\mathrel{\Big\vert}~}{\left\lvert {mK_S} \right\rvert}\neq \emptyset}\right\} \subseteq {\mathbf{Z}}\); this forms a semigroup. We will show that \({\mathcal{S}}\) is in fact a group.

We now discuss the proof of the theorem. Case 1: \(p_g=0, q=0, \chi({\mathcal{O}}_S) = 1\). By Riemann-Roch, \(\chi(D) = \chi({\mathcal{O}}_S) + {D(D-K)\over 2} = h^0(D) - h^1(D) + h^0(K-D)\), and so \(h^0(D) + h^0(K-D)\geq \chi(D)\). Applying this to \(D = mK\) yields \(h^0(mK) + h^0((1-m)K) \geq \chi({\mathcal{O}}_S)\). We’ll show a positive multiple and a negative multiple of \(K\) is effective, which can only happen if the multiple if zero. If \(\chi({\mathcal{O}}_S) = 1\), we have \(h^0(3K) + h^0(-2K)\geq 1\). If \(h^0(-2K) = 1\), then \(-2K \sim 0\). Consider the case where \(h^0(3K) \geq 1\). This implies \(p_3 = 1\). In this case we have \(h^0(2K)\geq 1\) since \(q=0\) and \(S\) is not rational. This implies \(p_2 = 1\), which implies \(p_1 = p_g = 1\), contradicting \(p_g=0\).

Case 2: we have already considered these in a previous chapter.

Case 3: \(p_g=1, q=0, \chi({\mathcal{O}}_S) = 2\), and we want to conclude \(K\sim 0\). By Riemann-Roch, \(h^0(2K) + h^0(-K) \geq 2\). We know \(h^0(2K) = 1\), thus \(h^0(-K)\geq 1\) and \(-K\) is effective. Since \(K\) is effective, we have \(K\sim 0\).

Case 4: \(p_g = 1, q=1, \chi({\mathcal{O}}_S) = 1\), we want to show this is impossible. Recall the SES \(\operatorname{Pic}^0(S)\hookrightarrow\operatorname{Pic}(S)\twoheadrightarrow{\operatorname{NS}}(S)\), and note \(\operatorname{Pic}^0(S) \cong {\mathbf{C}}^q/{\mathbf{Z}}^{2q} = {\mathbf{C}}/{\mathbf{Z}}^{2}\) in this case. This is a complex torus, which has a 2-torsion point \({\varepsilon}\in \operatorname{Pic}^0(S)\). By Riemann-Roch, \(h^0({\varepsilon}) + h^0(K-{\varepsilon}) \geq 1\). Since \({\varepsilon}\) can not be effective since \(2{\varepsilon}= 0\) (intersect with a hyperplane), we have \(h^0({\varepsilon}) = 0\). So \(h^0(K-{\varepsilon})\neq 0\). We thus have \(\exists D\in {\left\lvert {K} \right\rvert}\) and \(\exists E\in {\left\lvert {K-{\varepsilon}} \right\rvert}\). Note that \(2D=2E \in {\left\lvert {2K} \right\rvert}\) since there is a unique divisor in this linear system. Either this is zero, or effective and nonzero. We want to conclude \(D=E\), and thus \({\varepsilon}= D-E = 0\), a contradiction.

Called case 3.5 in class.

Case 5: the hardest case. Use the Albanese map \(\alpha: S\to \operatorname{Alb}(S)\), an abelian variety of dimension \(q=2\). Claim: \(\alpha\) is surjective and étale. Fact: if \(X\to Y\) is an etale map between projective varieties and \(Y\) is an abelian variety, then \(X\) is an abelian variety. I.e. any etale cover of an abelian variety is again an abelian variety. We have to prove the image of \(\alpha\) is not just a curve, and that \(\alpha\) is unramified.

Called case 4 in class.

Let \(S\) be a minimal surface with \(p_g = 1, q=0, \chi({\mathcal{O}}_S) = 2\) with \(\kappa = 0\). We proved \(K_S \sim 0\). By the Noether formula, \(\chi({\mathcal{O}}_S) = {K^2 + \chi(S)\over 12}\), so \(\chi(S) = 24\). We have the following Hodge diamond:

\begin{align*}\begin{tikzcd} && {h^{2,2}=1} \\ & {h^{1,2}=0} && {h^{2,1}=0} \\ {h^{0,2} = h^0(\omega_S) = 1} && {h^{1,1} = 20} && {h^{2,0}=1} \\ & {h^{0,1}=q=0} && {h^{1,0}=0} \\ && {h^{0,0} = 1} \end{tikzcd}\end{align*} > Link to Diagram

Each of these defines a K3 with an ample line bundle \(L \mathrel{\vcenter{:}}={\mathcal{O}}_X(-1)\), the restriction of \({\mathcal{O}}_{{\mathbf{P}}^n}(-1)\) to \(X\). In these cases:

• \(L^2 = H^2 X = 4\)
• \(L^2 = 2\cdot 3 = 6\)
• \(L^2 = 2\cdot 2\cdot 2 = 8\).

All of these yield a pair \((X, L)\), and we define the degree of the pair as \(d \mathrel{\vcenter{:}}= L^2\). These are the only complete intersections which are K3s.

By Riemann-Roch, \(\chi(D) = {D^2 - DK\over 2}\), and \(DK = 0\). This in fact holds for all minimal surfaces with \(\kappa = 0\), since some multiple of \(K\) is zero.

Suppose \(d=2\). Then \(L\) can not be ample. There is a double cover \(f:X\to {\mathbf{P}}^2\) branched over \(B = \left\{{f_6 = 0}\right\}\), a smooth curve given by an equation of degree 6. By Riemann-Hurwitz, \(K_X \sim_{\mathbf{Q}}f^*(K_{{\mathbf{P}}^2} + {1\over 2}B)\sim 0\), which forces \(B\) to be a sextic. Clearing denominators yields \(2K_X \sim f^*(2K_{{\mathbf{P}}^2} + B)\). We generally have \(f_* {\mathcal{O}}_X = {\mathcal{O}}_{{\mathbf{P}}^2} \oplus F^{-1}\) where \(F^2 = {\mathcal{O}}(B)\). This is the decomposition into the \(\pm 1\) eigenspaces of \(f_* {\mathcal{O}}_X\) respectively under an involution. There is a grading, this is given by a multiplication morphism \(F^{-1}\otimes F^{-1}\to {\mathcal{O}}_{{\mathbf{P}}^2}\), or equivalently a map \({\mathcal{O}}\to F^2\) where \(1\mapsto s\). The cover has local coordinates \(z^2 = s(x, y)\), and \((s) = B\) cuts out the ramification locus, so \(s\in {\mathcal{O}}_{{\mathbf{P}}^2}(B)\) and \(F^2 = {\mathcal{O}}_{{\mathbf{P}}^2}(B)\) is necessary. We have \(f_* \omega_X = \omega_{{\mathbf{P}}^2} \oplus \omega_{{\mathbf{P}}^2}(F)\). Moreover \(H^1({\mathcal{O}}_X) = H^1({\mathcal{O}}_{{\mathbf{P}}^2}) \oplus H^1({\mathcal{O}}_{{\mathbf{P}}^2}(-3)) = 0\), so \(q=0\). Similarly, \(H^0(\omega_X) = H^0({\mathcal{O}}_{{\mathbf{P}}^2}(-3)) \oplus H^0({\mathcal{O}}_{{\mathbf{P}}^2}(-3+3)) \cong 0 \oplus {\mathbf{C}}\), so \(\omega_X\) has a section and this forces \(K_X \sim 0\).

One could generalize this to consider double covers of other del Pezzo surfaces branched over \(B\in {\left\lvert {-2K} \right\rvert}\).

Minimal surfaces with \(\kappa = 1\) are elliptic, but not every elliptic surface satisfies \(\kappa = 1\). If the fibration admits a section, the fibers are genus 1 curves with a point and the fibration is called Jacobian. The fibers have Weierstrass form \(y^2 = x^3 + Ax + B\) where \(A(t_0, t_1), B(t_0, t_1)\) are homogeneous in the coordinates \(t_i\) on \({\mathbf{P}}^1\). One checks \(\Delta_{24} = 4A_8^3 + 27 B_{12}^2\).

Note \(L = f^* {\mathcal{O}}_{{\mathbf{P}}^1}(1)\) yields \(L^2 = 0\). One can choose other polarizations, e.g. \(L = s + df\). Then \(L^2 = s^2 + 2dsf + f^2 = -2 + 2d + 0 = 2d-2\), where \(s^2 = -2\) follows from the genus formula \(g(s) = {s^2 + sK\over 2} + 1\) with \(sK = 0\). For generic choices, \(L\) will be ample, and \(L^2 = 2d-2\) achieves every even integer.

Let’s count moduli for the examples given above.

• \(X_4\): there are \(7\choose 4\) monomials, minus \(\dim \operatorname{PGL}_4 = ?\), minus 1 for projectivizing equations. This yields \(35-16 = 19\).
• \(X_{2, 3}\): a more delicate count yields \(19\).
• \(X_{2, 2, 2}\): one gets 19 by computing the dimension of a Grassmannian.
• \(f: X\to {\mathbf{P}}^2\): the number of sextics in \({\mathbf{P}}^2\), minus \(\dim \operatorname{PGL}_2\), yields \({8\choose 2} - 9 = 19\).
• \(f:X\to {\mathbf{P}}^1\times {\mathbf{P}}^1\): counting yields 18.
• Elliptic: count possibilities for \(A_8\) and \(B_{12}\), modulo \(\operatorname{PGL}_2\), yields 18.

Why the difference: \(\operatorname{rank}\operatorname{Pic}(X_4) = 1\) generically, while \(\operatorname{rank}\operatorname{Pic}(X) = 2\) for e.g. \(X\to {\mathbf{P}}^1\times {\mathbf{P}}^1\). So larger Picard rank yields fewer moduli; in general one has \(20 - \operatorname{rank}\operatorname{Pic}(X)\) moduli.

In every even degree \(d\geq 4\), there exist K3s with a very ample line bundle. This comes from studying things like \(X_4\) which also admit an elliptic fibration, and setting \(\tilde L \mathrel{\vcenter{:}}= L + kF\) for \(F\) a fiber class. Then \(\tilde L^2 = L^2 + 2kLF\), and since \(L\) is very ample and \(F\) is basepoint free, \(\tilde L\) is again very ample.

Suppose \(X\) is a K3 and \(C \subseteq X\) is a smooth curve with \(g(C) \geq 2\). Note \(C^2 = 2g-2\) by the genus formula, since \(K=0\). Then \({\left\lvert {C} \right\rvert}\) is basepoint free, and \(\phi_{{\left\lvert {L} \right\rvert}}: X\to \operatorname{im}(X) \subseteq {\mathbf{P}}^3\) is degree either 1 or 2. In this case we say \(X\) is a hyperelliptic K3.

Next time: Hodge theory, deformations, Enriques surfaces. Last class: elliptic surfaces.

Note \(\rho \leq 20\) in characteristic zero, but possibly \(\rho = 22\) in positive characteristic.

K3 surfaces: \(X\) with \(p_g=1, q=0, K_X\sim 0, \chi(X) = 24, \chi({\mathcal{O}}_X) = 2\). Polarized: pairs \((X, L)\) with \(L\) an ample line bundle satisfying \(L^2 = 2d\). Note that any two K3 surfaces over \({\mathbf{C}}\) are homeomorphic, and \(\pi_1(X) = 1\) and thus \(H^1(X; {\mathbf{Z}}) = 0\). By UCT, \(H^2(X;{\mathbf{Z}}) = { \operatorname{Hom} }(H_1(X;{\mathbf{Z}}), {\mathbf{Z}}) \oplus T\), where \(T\) is the torsion in \(H_1\), is torsionfree is isomorphic to \({\mathbf{Z}}^{22}\). There is an integral bilinear form \(H^2(X;{\mathbf{Z}})\times H^2(X;{\mathbf{Z}})\to H^4(X;{\mathbf{Z}})\cong {\mathbf{Z}}\). Note that \((x, x)\) is even for all \(x\), symmetric, and unimodular. The form has signature \((3, 19)\). Such indefinite unimodular even forms are unique, so this lattice is \(II_{3, 19} \cong U^3 \oplus E_8^2\). Note that \(U = II_{1, 1}\) and \(E_8(-1) = II_{0, 8}\).

Note that \(\operatorname{Pic}(X) = H^1({\mathcal{O}}_X^{\times})\). Take the LES for the exponential SES \({\mathbf{Z}}\to {\mathcal{O}}_X\to {\mathcal{O}}_X^{\times}\) to get \begin{align*} 0\to \operatorname{Pic}^0(X) \to \operatorname{Pic}(X) \to {\operatorname{NS}}(X) = \ker\qty{H^2(X;{\mathbf{Z}})\xrightarrow{f} H^2({\mathcal{O}}_X)} .\end{align*} We have \(\operatorname{Pic}^0(X) = H^1({\mathcal{O}}_X)/H^1(X;{\mathbf{Z}}) = {\mathbf{C}}^q/{\mathbf{Z}}^{2q}\), and \(\operatorname{Pic}(X) \cong {\operatorname{NS}}(X) \subseteq H^2(X;{\mathbf{Z}}) = II_{3, 19}\). By the Hodge index theorem, \({\operatorname{NS}}(X)\) has signature \((1, r-1)\) for some \(r\). The inclusion \({\operatorname{NS}}(X) \subseteq H^2(X;{\mathbf{Z}})\) is a primitive embedding of lattices, i.e. \(nv\in {\operatorname{NS}}(X)\implies v\in {\operatorname{NS}}(X)\), since \({\operatorname{NS}}(X)\) is isomorphic to a kernel. Nikulin classified which hyperbolic lattices can be primitively embedded into \(II_{3, 19}\), and moreover shows they are all attained by some K3 surface.

Understanding \(\ker f\) from above: it is the composition \begin{align*} H^2(X;{\mathbf{Z}}) \to H^2(X;{\mathbf{C}}) = H^{2, 0} \oplus H^{1, 1} \oplus H^{0, 2} \twoheadrightarrow H^{2, 0} = H^2({\mathcal{O}}_X) .\end{align*} Write \(H^{0, 2} = H^0(\Omega^2) = H^0(K_X) = {\mathbf{C}}\omega_X\).

Thus varying \(\omega_X\) amounts to varying \({\operatorname{NS}}(X)\), and one has \(0\leq r\leq 20\) for \(r \mathrel{\vcenter{:}}=\operatorname{rank}{\operatorname{NS}}(X)\). We define a period domain \begin{align*} D \mathrel{\vcenter{:}}=\left\{{[\omega] {~\mathrel{\Big\vert}~}\omega \in II_{3, 19} \otimes{\mathbf{C}}, \omega\cdot\omega = 0, \omega\cdot \overline{\omega }> 0}\right\} \subseteq {\mathbf{CP}}^{21} .\end{align*} The Torelli theorem states that for any \(\omega\in D\) there exists a complex analytic K3 surface \(X\) with \(\omega_X = \omega\), together with a marking \(H^2(X;{\mathbf{Z}}) { \, \xrightarrow{\sim}\, }II_{3, 19}\). Further, two marked K3s are isomorphic iff \(\omega_X = \omega_{X'}\).

Multiple fibers are allowed, as well as non-reduced and non-irreducible curves.

Such surfaces \((X, E)\) with a section biject with del Pezzo surfaces of degree 1. Start with such a \(X = \dP_1\) and consider \({\left\lvert {-K_X} \right\rvert}\). Since \(h^0(-K_X) = 2\), this linear system is a \({\mathbf{P}}^1\) and any two elements intersect at one point. This yields a pencil with a basepoint. If \(C\in {\left\lvert {-K_X} \right\rvert}\), then \(K_C = K_X+C\mathrel{\Big|}_C = 0\), so this is an elliptic curve. Conversely, taking an RES and blowing down \(E\) yields a \(\dP_1\).

Such a family of curves can be regarded as a single curve over the function field of \({\mathbf{P}}^1\). There is still a section, so Mordell’s theorem guarantees that the sections form a finitely generated group. This produces a surface with infinitely many \((-1)\)-curves.

All surfaces \(S\) with \(\kappa(S) = 1\) are elliptic. The fibration is induced by \(mK_X\) for \(m \gg 1\). The rational map it induces maps to a curve since \(\kappa = 1\). The nontrivial fact is that this map is basepoint free and is thus regular. Adjunction shows that the generic fiber is an elliptic curve.

\(K_X.C \geq 0\) for all curves \(C\). However, the restriction of \(K_X\) to a fiber must be zero, so \(K_X.F . F = 0\) for every fiber \(F\). So every multiple \(mK_X\) must contract every fiber, along with all curves it intersects by zero. Thus the image must be a curve, but \(\kappa = 2\) forces the image to be a surface.

A generic K3 has \(\rho = 1\), but any elliptic fibration has a fiber and a multisection, so \(\rho \geq 2\). So the generic K3 is not elliptic.

Halphen pencils: start with a cubic \(f_3\) and a sextic \(g_6\) and take the pencil \(sf_3 + tg_6\). The intersection \(f_3 \cap g_6\) is 9 points, and we can arrange for the multiplicity is 2 at each point. Taking \(\operatorname{Bl}_9 {\mathbf{P}}^2\), this has a fibration to \({\mathbf{P}}^1\). This is a RES without a section with a multiple fiber.

There is a variation \(sf_{3d} + tg_{3d}\), and blowing up again yields a RES with multiple fiber. We recover the previous case by taking \(f_{3d} = f_3^d\) for \(d=2\).

There is a construction with takes elliptic surfaces \(\pi: X\to C\) to Jacobian elliptic surfaces \(:\pi: JX\to C\), i.e. elliptic surfaces with a section. On smooth fibers, this sends \(X_t\) to \((JX_t, 0)\), since the Jacobian has a distinguished origin. This construction removes multiple fibers, and on singular fibers one takes a compactified Jacobian. Multiple curves \(nE\) are mapped to \(E\), any other curves \(X_t\) are sent to \(JX_t\cong X_t\) (which has a section). It follows that \(\chi(X_t) = \chi(JX_t)\).

What happens to an Enriques surface under this construction? Considering the first example: since \(\chi({\mathbf{P}}^2) = 3\) and \(\chi({\mathbf{P}}^1) = 1\), we have \(\chi(\operatorname{Bl}_9 {\mathbf{P}}^2) = 12\) since points are replaced with \({\mathbf{P}}^1\)s, increasing \(\chi\) by one each time. Note that \(\chi({\mathcal{O}}_X) = 1\) for this RES, for K3s \(\chi({\mathcal{O}}_X) = 2, \chi = 24\), and for Enriques surfaces \(\chi({\mathcal{O}}_X) = 1, \chi = 12\). Here we’ve used that \(K^2 = 0\) and thus \(\chi = 12 \chi({\mathcal{O}}_X)\) by Noether’s formula.

Since \(g(E) = 1\), the singular fibers are either of the for \(mE\), or a cuspidal or nodal curve. If the singular fiber is a collection of \((-1)\)-curves, take the dual graph. This has an associated matrix \(M_{ij} = C_i.C_j\) which is negative semidefinite. The possible graphs that can arise are \(\tilde A_n, \tilde D_n, \tilde E_m\) for \(m=6,7,8\). Taking the determinants of these matrices yields

• \(\operatorname{det}A_n = n+1\),
• \(\operatorname{det}D_4 = 4\),
• \(\operatorname{det}E_6 = 3\),
• \(\operatorname{det}E_7 = 2\),
• \(\operatorname{det}E_8 = 1\).

These graphs uniquely recover the fibers, with the exception of \(\tilde A_0,\tilde A_1, \tilde A_2\).

One has \(M(I_n) = { \begin{bmatrix} {1} & {n} \\ {0} & {1} \end{bmatrix} }\) and \(M(I_n^*) = { \begin{bmatrix} {1} & {-n} \\ {0} & {1} \end{bmatrix} }\) for monodromy around these fibers.

The canonical class formula: \begin{align*} K_X \sim_{\mathbf{Q}}\pi^* K_C + \sum {m_i - 1\over m_i} F_i + M .\end{align*} where the sum is over multiple fibers and called the divisor part, and \(M\) is called the moduli part. One sets \(M = j^* {\mathcal{O}}_{{\mathbf{P}}^1_j}(1) + \sum c_i F_i\) where the \(c_i\) are prescribed based on the Dynkin type. Here \(j: C\to {\mathbf{P}}^1\) is the map sending a fiber \(X_t\) for \(t\in C\) to its j-invariant. This puts significant restrictions on the fiber types.