# 2024-11-26-12-46-18: Elliptic Surfaces ## Examples of Elliptic Surfaces :::{.definition} An **elliptic surface** $X$ is a (smooth) fibration $\pi: X\to C$ over a curve $C$ whose generic fiber is a genus 1 curve, not necessarily with an origin fixed. We assume there are no $(-1)$-curves in a fiber, so $\pi$ is relatively minimal. ::: :::{.remark} Multiple fibers are allowed, as well as non-reduced and non-irreducible curves. ::: :::{.example} Begin with $\PP^2$ and take two general polynomials $f_3(x_0, x_1, x_2), g_3(x_0, x_1, x_2)$ and consider the pencil $sf_3 + tg_3 = 0$ where $[s:t] \in \PP^1$. Generically, $f_3 \intersect g_3$ is 9 points, so blow these up. These yields a pencil without basepoints with a map $\Bl_9 \PP^2 \to \PP^1$ with coordinates $[s: t]$. The generic fiber is an elliptic curve, and this surface has 9 sections $E_1, \cdots, E_9$. More generally, it's enough to assume this pencil contains an irreducible curve, and the points of blowup are allowed to be infinitely near (e.g. tangency of the curves). In this case, at least $E_9$ is a section, so this always yields an elliptic surface with a section. Thus the points need not be in general position. The most degenerate case is when all other 8 curves are contained in a fiber. In this case, $\kappa = - \infty$, so $X$ is a RES (rational elliptic surface). Note that there are many $(-1)$-curves, but they are all sections and not contained in fibers. ::: :::{.remark} Such surfaces $(X, E)$ with a section biject with del Pezzo surfaces of degree 1. Start with such a $X = \dP_1$ and consider $\abs{-K_X}$. Since $h^0(-K_X) = 2$, this linear system is a $\PP^1$ and any two elements intersect at one point. This yields a pencil with a basepoint. If $C\in \abs{-K_X}$, then $K_C = K_X+C\mid_C = 0$, so this is an elliptic curve. Conversely, taking an RES and blowing down $E$ yields a $\dP_1$. ::: :::{.remark} Such a family of curves can be regarded as a single curve over the function field of $\PP^1$. 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. ::: :::{.example} Some K3 surfaces are elliptic; there is an 18-dimensional family of them. Produce these using Weierstrass equations. Here $\kappa = 0$. ::: :::{.example} All Enriques surfaces are elliptic (without a section). There are always two double fibers in a fibration over $\PP^1$. Note that if you have a section, you can not have a multiple fiber, by considering intersection multiplicities. Again, here $\kappa = 0$. ::: :::{.remark} 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. ::: :::{.lemma} Surfaces with $\kappa(S) = 2$ are not elliptic. ::: :::{.remark} $K_X.C \geq 0$ 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. ::: :::{.remark} 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. ::: :::{.remark} Halphen pencils: start with a cubic $f_3$ and a sextic $g_6$ and take the pencil $sf_3 + tg_6$. The intersection $f_3 \intersect g_6$ is 9 points, and we can arrange for the multiplicity is 2 at each point. Taking $\Bl_9 \PP^2$, this has a fibration to $\PP^1$. 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 $d=2$. ::: :::{.remark} 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 $\chi(X_t) = \chi(JX_t)$. ::: :::{.remark} What happens to an Enriques surface under this construction? Considering the first example: since $\chi(\PP^2) = 3$ and $\chi(\PP^1) = 1$, we have $\chi(\Bl_9 \PP^2) = 12$ since points are replaced with $\PP^1$s, increasing $\chi$ by one each time. Note that $\chi(\OO_X) = 1$ for this RES, for K3s $\chi(\OO_X) = 2, \chi = 24$, and for Enriques surfaces $\chi(\OO_X) = 1, \chi = 12$. Here we've used that $K^2 = 0$ and thus $\chi = 12 \chi(\OO_X)$ by Noether's formula. ::: :::{.lemma} Let $X$ be relatively minimal. Then $K_X \sim cF$ where $c\geq 1$ and $F$ is a fiber. ::: :::{.proof} Write $K_X = \sum n_i C_i$ with $C_i$ in fibers. Pick a fiber $F$, then for some index set $J$ we have $\sum m_j C_j = F$. Compute $p_a(F) = 1$ since this is constant in a flat family. On the other hand, we have $p_a(F) = {(K_X+F)F\over 2} + 1$, and since $F^2 = 0$ we have $K_X.F = 0$. Then either 1. $K_X.C_j = 0$ for all $j$, or 2. $\exists j$ such that $K_x C_j < 0$. This follows from the fact that $K_X.F = \sum m_j K_X.C_j = 0$. If $C_j^2 < 0$, then note that $p_a(C_j) = {K_X.C_j + C_j^2 \over 2} + 1$. Then either - $F = m_j C_j$ is irreducible, or - $p_a(C_j) = 0, C_j \cong \PP^1, C_j^2 = -2$, and $K_X.C_j = 0$. This follows from the fact that the numerator is either $(-1) + (-1)$, which is ruled out by relative minimality, or $(0) + (-2)$. The latter is the only possibility. In any case, $K_X.C_j = 0$, and $\sum n_i C_i = \sum d_i F_i$ for some collection of fibers $F_i$. ::: ## Singular fibers :::{.remark} 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 - $\det A_n = n+1$, - $\det D_4 = 4$, - $\det E_6 = 3$, - $\det E_7 = 2$, - $\det E_8 = 1$. These graphs uniquely recover the fibers, with the exception of $\tilde A_0,\tilde A_1, \tilde A_2$. ![](figures/2024-11-26_13-56-02.png) One has $M(I_n) = \matt 1 n 0 1$ and $M(I_n^*) = \matt{1}{-n}{0}{1}$ for monodromy around these fibers. ::: :::{.remark} The canonical class formula: \[ K_X \sim_\QQ \pi^* K_C + \sum {m_i - 1\over m_i} F_i + M .\] where the sum is over multiple fibers and called the *divisor part*, and $M$ is called the *moduli part*. One sets $M = j^* \OO_{\PP^1_j}(1) + \sum c_i F_i$ where the $c_i$ are prescribed based on the Dynkin type. Here $j: C\to \PP^1$ is the map sending a fiber $X_t$ for $t\in C$ to its j-invariant. This puts significant restrictions on the fiber types. :::