# 2024-08-15-12-47-25 Introduction Main topic: algebraic surfaces, always assumed projective. Can we classify all such surfaces? Either - Up to isomorphism (biregular classification) - Up to birational isomorphism (birational classification, weaker) Recall that $X_1 \birational X_2$ iff they share a Zariski open subset, noting that open sets are dense. Equivalently, there is an isomorphism of function fields $k(X_1) \cong k(X_2)$, so varieties up to birational isomorphism are equivalent to finitely generated field extensions. We will take $k=\kbar$ characteristic zero, and assume $k=\CC$ and apply the Lefschetz principle -- any such field admits an embedding $k\injects \CC$. Fields of other characteristics will appear. Note that $\dim X = d$ corresponds to fields of transcendence degree $d$ over $k$. For $d=1$, we are considering smooth projective curves. However, $C_1\cong C_2 \iff C_1 \birational C_2$, so there is no birational geometry to speak of in this dimension. The basic invariant is the genus $g(C) \geq 0$. - If $g=0$ then $C\cong \PP^1$, - if $g=1$ then $C$ is an elliptic curve. - If $g\geq 2$ (which includes most curves), there exists a moduli space $\Mg$ of such curves. Points of $\Mg$ correspond to isomorphism classes of such curve -- there are infinitely many such classes, but they are organized into a variety. This space is almost smooth but is not complete. There exists a Deligne-Mumford compactification $\Mgbar$, the boundary curves are so-called *stable curves*. ## Basic tools We will introduce tools which will be black-boxed in order to apply them to the classification problem. For $S$ a surface, divisors $D$ are formal linear combinations of curves. For any divisor $D$, there is an invertible (rank 1) locally free sheaf $\OO_S(D)$. The sections are locally regular functions. Divisors modulo linear equivalence correspond to invertible sheaves up to isomorphism, we call this group $\Pic(S)$. We introduce numerical invariants to generalize the genus. The first is the irregularity $q = h^0(\Omega_S^1)$. For curves, one recovers $q = g$. Another is the geometric genus $p_g = h^0(\det \Omega_S) = h^0(\omega_S)$, the dimension of the space of sections of the top-degree differentials. Note that $\omega_S \cong \OO_S(K_S)$ for a canonical divisor $K_S$ on $S$. For curves, $\deg K_S = 2g-2$. This splits the classification into three cases: - $g=0 \iff K_S < 0 \iff \kappa = -\infty$, - $g=1 \iff K_S = 0 \iff \kappa = 0$, - $g\geq 2 \iff K_S > 0 \iff \kappa = 1$. This is the *general type* case. The plurigenus is defined as $p_m = h^0(mK_S)$. As $m\to \infty$, it is a fact that $p_m \sim m^k$ for some $k$, we define $\kappa \da k$. One generally has $\kappa \in \ts{-\infty,1,2}$ for surfaces, where $\kappa = -\infty$ iff $p_m = 0$ for all $m \gg 0$. The classification of surfaces similarly is by Kodaira dimension. ## Birational geometry of surfaces Let $p\in S$ be a point of a smooth surface. Consider $\Bl_p S$, which replaces $p$ with $E\cong \PP^1$ and $E^2 = -1$. To define the intersection pairing -- look at the normal bundle $N_{E/S} = \OO_S(-1)$. There is a map $\Bl_p(S) \to S$ which is an isomorphism away from $p$ and $E$. Castelnuovo's criterion gives a converse: any such curve can be blown down. Note that $E$ is referred to as an exceptional curve of the first kind, i.e. a $(-1)$-curve. Exceptional curves of higher kinds are rarely used. Fact: any birational isomorphism of surfaces factors as a composition of blowups and blowdowns. We say $S$ is *minimal* if there are no $(-1)$-curves. Otherwise just blow them down, i.e. contract those curves. We will introduce a numerical invariant that decreases under blowdowns, so this process terminates. Recall that there is an intersection product $C_1C_2$ which, if they intersect transversally, counts intersection points. In more singular situations, it takes into account intersection multiplicity. Note that everything is oriented when over $\CC$, so $C_1 C_2 \geq 0$ for any curves. How does one compute $E^2$? Replace one copy of $E$ by $D_1 - D_2$ and compute $E(D_1 - D_2)$ instead. New goal: classify minimal surfaces. - $\kappa = -\infty$: $S = \PP^2$ or geometrically ruled surfaces (fibrations over a curve $C$ with $\PP^1$ fibres). If $C=\PP^1$ in this fibration, one obtains a Hirzebruch surface $F_n$ for $n\geq 0$ and $n\neq 1$ (which is not minimal). These are rational surfaces. Note that "minimal" is classical terminology and is no longer used for $\kappa = -\infty$: these are instead called Mori Fano fibrations. - $\kappa = 0$: abelian surfaces and K3 surfaces ($K_S = 0$), Enriques surfaces ($K_S\neq 0, 2K_S = 0$) and $p_g=q=0$, or bielliptic surfaces (products $E_1\times E_2/\ZZ_n$ where $n=2,3,4,6$ and $E_i$ are elliptic curves). - $\kappa = 1$: elliptic surfaces (admit fibrations over curves whose general fiber is an elliptic curve). - $\kappa = 2$: general type (most surfaces). Note that if $\kappa \geq 0$ and $S$ is minimal then $K_S$ is nef, i.e. $K_S \cdot C \geq 0$ for any curve $C$. One can take this as the definition of minimal. Note that one has $K_S = mF$. If $m < 0$ then $C\cong \PP^1$, if $m=0$ then $S$ is K3 or Enriques, and if $m > 0$ then $\kappa = 1$. The general type surfaces are hard to classify, except for special cases. E.g. $p_g = q = 0$ which give counterexamples to these invariants determining rationality. Examples include Godeaux, Campadelli, Balrov, Inuoe, Burniat surfaces. These are sometimes referred to as fake projective planes, famously there are 100 of these. Note Castelnuovo's criterion: $S$ is rational iff $p_2 = q = 0$. One can consider the moduli of surfaces of general type $M_{c_1^2, c_2}$ where $c_1^2 = K_S^2$ and $c_2$ is another numerical invariant. These are sometimes called Gieseker moduli space. There are KSBA compactifications, since GIT does not work well in the setting of surfaces. ## Other subjects We can discuss: - K3s - Characteristic $p$, mainly $p=2,3$ - Complex-analytic surfaces (non-algebraic), algebraic dimension $a\in \ts{0,1,2}$. Here $a=2$ is the algebraic case, and the other cases are non-algebraic. - Non-closed fields $k\neq \kbar$ - Singular surfaces Why studying smooth surfaces suffices: any variety admits a normalization, and any normal surface has a unique minimal resolution in any characteristic. In higher dimensions, existence of resolutions is generally an open problem, and are almost never unique. Thus birational geometry for threefolds is significantly harder.