# Thursday, March 16 :::{.definition title="Linear systems"} For $X$ smooth projective, $L\in \Pic(X)$, $V \subseteq H^0(L)$, define $\PP V$ to be a linear system (a collection of linearly equivalent divisors) and $\abs{L} \da \PP H^0(L)$ to be a complete linear system. If $s\in H^0(L)\smz$, then $V(s)\in \Div(X)$ and $\OO(V(s)) = L$ (noting that all sections are linearly equivalent). Here we projectivize since $V( \lambda s) = V(s)$. ::: :::{.example title="?"} Let $X = \PP^1$ and $L = \OO(1)$, then \[ H^0(L) = \ts{ f\in k[x,y] \st f \text{ homogenous },\, \deg f = n} .\] As divisors, \[ \PP H^0(\OO(1)) = \ts{\sum a_i p_i \st a_i > 0,\, \sum a_i = n} ,\] corresponding to the zeros and multiplicities of $f$. ::: :::{.example title="?"} Let $E = \CC / \Lambda$ be elliptic and $L = \OO(D)$ for $D = 3[0]$. Then $\abs{L} = \ts{ [p] + [q] + [r] \st p+q+r =0 \mod \Lambda }$. Note that $\abs{L} \cong \PP^2$ since $r$ is determined by $p, q$. ::: :::{.example title="?"} Let $C$ be a curve with $g\geq 2$ and $L = K_C = \Omega^1_C$. Then by RR $h^0(K_C) = g$ and $\abs{K_C} \cong \PP^{g-1}$ is called the **canonical linear system**. ::: :::{.definition title="?"} Let $V \leq H^0(L)$ be a subspace and $\ts{s_0, \cdots, s_k}$ be a basis. Then there is a map \[ \phi_V: X &\rational \PP^k \\ x &\mapsto [s_0(x):\cdots : s_k(x)] .\] Defining the **base locus** as $\Bs(L) \da \ts{x\in X\st s_i(x) = 0 \, \forall i}$, note $\phi_{\PP V}$ is not well-defined for any $x\in \Bs(L)$. ::: :::{.proposition title="?"} For $C$ a curve of $g\geq 2$, $\Bs(K_X) = \emptyset$. ::: :::{.proof title="?"} STS $\forall p\in C$ there is some $s\in H^0(K_C)$ with $s(p) \neq 0$. Letting $s_p$ be a section vanishing only at $p$, multiplication by $s_p$ induces $H^0(K_C(-p)) \injects H^0(K_C)$ with image the sections of $K_C$ vanishing at $p$. Thus STS this is not surjective by showing $h^0(K_C) > h^0(K_C(-p))$. Apply RR and Serre duality: \[ h^0(K_C(-p)) - h^1(K_C(-p)) &= h^0(K_C(-p)) - h^0(\OO(p))\\ &= \deg K_C(-p) + (1-g)\\ &= (2g-3) + (1-g)\\ &= g-2 .\] Note that if $s\in H^0(\OO(p))$ then $V(s) = [q]$ must be a single point, but if $p\neq q$ then $[p]-[q] = 0$ and $\exists f: C\to \PP^1$ with $f\inv(0) = p$ and $f\inv(\infty) = q$ with $\deg f = 1$, forcing $C \cong \PP^1$ and contradicting $g \geq 2$. So $p=q$, and $h^0(\OO(p)) = 1$, and thus \[ h^0(K_C(-p)) - 1 = g-2 \implies h^0(K_C(-p)) = g - 1 < g = h^0(K_C) .\] ::: :::{.exercise title="?"} Show that if $C$ is not hyperelliptic ($\exists f: C\to \PP^1$ 2-to-1) then $\forall p,q\in C$ one can find $s \in H^0(K_C)$ with $s(p) = 0, s(q)\neq 0$, so they are separated by linear forms on $\PP^{g-1}$. This yields an actual morphism $\phi_{\abs{K_C}}: C\to \PP^{g-1}$ where $p, q$ are not mapped to the same point. This is the canonical embedding of a curve, which only works when $g\geq 3$ and $C$ is non-hyperelliptic. If $g = 2$ or $C$ is hyperelliptic, $3K_C$ yields an (tricanonical) embedding. ::: :::{.example title="?"} If $C$ is not hyperelliptic and $g = 3$, then $C\embeds \PP^2$ by the canonical embedding. This yields an element in $\Div(\PP^2)$, which is a smooth quartic. ::: :::{.proposition title="Canonical of $\PP^n$."} \[ K_{\PP^n} = \OO(-n-1) .\] ::: :::{.proof title="?"} Take coordinates $[x_0:\cdots : x_n]$ and take $\omega \da {\dx_1\over x_1} \wedge \cdots \wedge {\dx_n\over x_n}$, noting that this omits $x_0$. This has poles along each $V(x_i)$ for $i\neq 0$ and in fact a pole at $x_0$, since these are $n+1$ distinct spaces. E.g. for $n=1$, since $x_1 = x_0\inv$ we have ${ d(x\inv)\over x\inv } = - {\dx_1\over x_1}$. ::: :::{.proposition title="Adjunction"} If $X$ is smooth and $D\in \Div(X)$ then \[ K_D = \ro{\qty{K_X \tensor \OO(D)}}{D} .\] ::: :::{.proof title="?"} Omitted, take residues. ::: :::{.remark} Applying this to the previous curve situation: note $\deg K_C = 2g-2 = 4$, which counts $\phi_{\abs{K_C}}(C) \intersect V(x_0)$ and yields a quartic. ::: :::{.proposition title="?"} For $C$ a degree $d$ curve in $\PP^2$, \[ g = {d-1\choose 2} .\] ::: :::{.proof title="?"} Take degrees in the adjunction formula and apply Bezout's theorem: \[ K_{C} = K_{\PP^2} + C \mid_C \implies 2g(C) - 2 = = \deg(-3H + dH)\mid_C = d (d-3) .\] More generally if $C \subseteq S$ a curve in a surface, $\deg L \mid_C = c_1(L) . [C]$ is an intersection number. Expanding this yields $g = {d^2 - 3d +2 \over 2} = {d-1\choose 2}$. ::: :::{.definition title="Kodaira dimension"} \[ \kappa(X) \da \max_{n > 0} \ts{\dim \im \phi_{nK_X} } \] where $\dim \emptyset \da -\infty$ and $\kappa(X) \in \ts{-\infty,0,\cdots, \dim X}$. ::: :::{.remark} Note that since $3K_C$ yields an embedding for a curve with $g\geq 2$, $\kappa(C) = 1$. Also note that $\kappa(X) = -\infty \iff h^0(nK_X) = 0$ for all $n$. For curves: | $g(C)$ | $\kappa(C)$ | |--------|-------------| | $0$ | $-\infty$ | | $1$ | $0$ | | $2$ | $1$ | | $3$ | $1$ | | $4$ | $\vdots$ | Here we've used that $K_{\PP^1} = \OO(-2)$ has no sections and $K_E = \OO_E$ is trivial. Fanos are $\kappa = -\infty$ and general type ($\kappa(X) = \dim X$) are $g\geq 2$. ::: :::{.remark} For smooth projective surfaces: $\kappa(S) \in \ts{ -\infty,0,1,2 }$. See Beauville for the Enriques-Kodaira classification due to the Italian school: - $\kappa(S) = -\infty$: ruled and rational. - $\kappa(S) = 0$: K3, Abelian, Enriques, Bi-elliptic. - $\kappa(S) = 1$: Elliptic. - $\kappa(S) = 2$: General type. ::: :::{.definition title="?"} $S$ is **ruled** if $\exists \pi: S\to C$ with generic fiber $\cong \PP^1$: ![](figures/2023-03-16_10-50-54.png) $S$ is **rational** if $S\birational \PP^2$. :::