--- title: "Flashcards Enriques Surfaces" aliases: ["Flashcards Enriques Surfaces"] flashcard: "Reading::EnriquesProject" created: 2023-03-26T11:58 updated: 2023-04-11T10:50 --- # Flashcards Surfaces ## General Surfaces ## Definitions - [ ] What is an smooth Artin stack? - [ ] What is a unirational stack? - [ ] What is a pencil of curves? - [ ] What is a 2-reflective lattice? - [ ] What is the radical of a quadratic space? - [ ] What is a Cremona transformation? - [ ] What is a Prym cuve? ## Standard Theorems - [ ] What is the Hodge index theorem? - [ ] What is the Lefschetz theorem? - [x] What is Vinberg's algorithm? ✅ 2023-03-30 - [ ] Given a 2-reflective lattice, computes a fundamental domain. ### Standard invariants - [x] What is the irregularity of a surface? ✅ 2023-03-30 - [ ] $q \da \dim \Pic(X) = \dim \Alb(X)$, over $\CC$ one has $q = h^{0, 1} = \dim H^1(\OO_X)$. - [ ] $q = p_g - p_a$ - [ ] For smooth Kahlers, coincides with $h^{1, 0}$, the number of global 1-forms. - [x] What is the geometric genus of a surface? ✅ 2023-03-30 - [ ] $p_g \da h^{2, 0} = P_1$ the number of volume forms. - [ ] More generally $p_g = h^{\dim X, 0} = \dim H^0(\det \Omega_X)$: ![](2023-03-30-1.png) - [x] What is the arithmetic genus of a surface? ✅ 2023-03-30 - [ ] $p_a \da p_g - q = h^{0, 2} - h^{0, 1}$. - [ ] More generally, $p_a=(-1)^r\left(\chi\left(\mathcal{O}_X\right)-1\right)$. - [ ] For smooth projective, $p_a = \sum_j (-1)^j h^{n-j, 0}$: ![](2023-03-30.png) - [x] What does it mean for $K_X$ t o be nef? ✅ 2023-03-30 - [ ] $K_X . D \geq 0$ for all $D\in \Div(X)^\eff$. - [ ] What does $\Pic(X)$ parameterize? - [ ] Divisor classes modulo linear equivalence. - [x] What does $\Pic^0(X)$ parameterize? ✅ 2023-03-30 - [ ] Divisor classes algebraically equivalent to zero (a clopen subscheme). - [x] What does $\Pic^\tau(X)$ parameterize? ✅ 2023-03-30 - [ ] Divisor classes **numerically** equivalent to zero (a clopen subscheme). - [x] What is $\Num(X)$? ✅ 2023-03-30 - [ ] $\Pic(X)/\Pic^\tau(X) \cong \NS(X)/\tors$. ### Types of Surfaces - [x] What is an Enriques surface? ✅ 2023-03-30 - [ ] $q=0$ with $K_X\neq 0$ but $2K_X = 0$. - [ ] All arise as $\K3/\iota$ for a fixed-point-free involution. - [ ] What is an abelian surface? - [x] What is a K3 surface? ✅ 2023-03-30 - [ ] $\kappa = 0$, $q=0$, and $K_X = 0$. - [x] What is a Kummer surface? ✅ 2023-03-30 - [ ] $\Bl_{16}(A/\iota)$. - [ ] What is a class $\rm{VII}$ surface? - [ ] What is a type $\rm{VII}_0$ surface? - [ ] What is an Inoue surface? - [ ] What is a Hopf surface? - [ ] What is a del Pezzo surface? - [ ] What is a Fano surface? - [ ] What is a Veronese surface? - [ ] What is an Inoue-Hirzebruch surface? - [ ] What is an Enoki surface? - [ ] What is a Kato surface? - [ ] What is a Kahler surface? - [ ] Three compatible structures $(X, J, g, \omega)$: (almost) complex, Riemannian, symplectic. - [ ] Equivalently, $(X, h)$ a Hermitian manifold where the associated 2-form $\omega \in H^{1, 1}$ is closed. - [ ] Equivalently, $\Hol X \leq \U_n$. - [ ] Equivalently, $b_2 \in 2\ZZ$. - [ ] What is a Kodaira surface? - [ ] What is a primary Kodaira surface? - [ ] What is a secondary Kodaira surface? - [ ] What is a bielliptic surface? - [ ] What is a hyperelliptic surface? - [ ] What is a quasi-hyperelliptic surface? - [x] What is an elliptic surface? ✅ 2023-03-30 - [ ] A surface equipped with an elliptic fibration. - [x] What is an elliptic fibration? ✅ 2023-03-30 - [ ] Surjective holomorphic map to a curve where all but finitely many fibers are smooth irreducible curves with $g=1$. - [x] What is a quasi-elliptic surface? ✅ 2023-03-30 - [ ] Almost elliptic, but almost all fibers are allowed to be degnerate elliptic curves: rational curves with a single node. - [ ] What is a Hilbert modular surface? - [ ] What is a fake projective plane? - [ ] What is a Barlow surface? - [ ] What is a unirational surface? - [ ] What is a supersingular K3 surface? - [ ] What is a Horikawa surface? - [ ] What is a minimal surface? ## Definitions: Singularities - [ ] What is an ordinary double point? Triple? - [ ] What is a tacnode? ## Hodge Diamonds - [x] What is the Hodge diamond of $\PP^2$? ✅ 2023-03-30 - [ ] ![](2023-03-30-2.png) - [x] What is the Hodge diamond of $F_n$ the $n$th Hirzebruch surface? ✅ 2023-03-30 - [ ] ![](2023-03-30-3.png) - [x] What is the Hodge diamond of a ruled surface? ✅ 2023-03-30 - [ ] ![](2023-03-30-4.png) - [x] What is the Hodge diamond of a class $\rm{VII}$ surface? ✅ 2023-03-30 - [ ] ![](2023-03-30-5.png) - [x] What is the Hodge diamond of a K3 surface? ✅ 2023-03-30 - [ ] ![](2023-03-30-6.png) - [x] What is the Hodge diamond of an abelian surface? ✅ 2023-03-30 - [ ] ![](2023-03-30-7.png) - [x] What is the Hodge diamond of an Enriques surface? ✅ 2023-03-30 - [ ] ![](2023-03-30-8.png) - [x] What is the Hodge diamond of a hyperelliptic/bielliptic surface? ✅ 2023-03-30 - [ ] ![](2023-03-30-9.png) ## Classification - [x] Which surfaces have $\kappa(X) = -\infty$? ✅ 2023-03-30 - [ ] Rational $(q=0)$ or ruled $(q\gt 0)$, or the non-algebraic non-Kahler class $\rm{VII}$ surfaces (Hopf or Inoue). - [x] What is the birational classification of minimal rational surfaces? ✅ 2023-03-30 - [ ] $\PP^2$ or $F_n$ a Hirzebruch surfacefor $n=0$ corresponding to $(\PP^1)^2$ or $n\geq 2$ corresponding to $\Tot(\OO(1)\sumpower{2})$. - [x] What surfaces have $\kappa(X) = 0$? ✅ 2023-03-30 - K3, - Abelian surfaces and complex tori, - Kodaira surfaces - Enriques surfaces - Hyperelliptic/bielliptic surfaces - [x] What surfaces have $\kappa(X) = 1$? ✅ 2023-03-30 - [ ] Always elliptic - [ ] Possibily quasi-elliptic in characteristic 2 or 3. - [ ] Beware: $\kappa(X) \implies$ elliptic but not conversely! E.g. Enriques are elliptic. - [x] What is the BMY (Bogomolov–Miyaoka–Yau) inequality? ✅ 2023-03-30 - [ ] $c_1^2 \leq 3c_2$, for surfaces of general type. - [x] What is the Noether inequality? ✅ 2023-03-30 - [ ] $5 c_1^2-c_2+36 \geq 0$ ## Formulas - [x] What is Noether's formula for surfaces? ✅ 2023-03-30 - [ ] $12\left(1-q+p_g\right)=K_S^2+c_2$ where $c_2=\sum_k (-1)^k \beta_k(S) = \chi(S)$ over $\CC$. ## Fibers - [x] What is an $\rm{I}_0$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-10.png) - [x] What is an $\rm{I}_1$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-11.png) - [x] What is an $\rm{I}_2$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-12.png) - [x] What is an $\rm{I}_n$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-13.png) - [x] What is a $\rm{II}$ fiber? ✅ 2023-03-30 - [ ] A cusp: ![](2023-03-30-14.png) - [x] What is a $\rm{III}$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-15.png) - [x] What is an $\rm{IV}$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-16.png) - [x] What is an $\rm{I}_0^*$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-17.png) - [x] What is an $\rm{I}_n^*$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-18.png) - [x] What is an $\rm{IV}^*$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-19.png) - [x] What is an $\rm{III}^*$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-20.png) - [x] What is an $\rm{II}^*$ fiber? ✅ 2023-03-30 - [ ] ![](2023-03-30-21.png) ## K3 Surfaces - [x] What is the nowhere vanishing holomorphic form on the Fermat quartic $F_4: x^4+y^4+z^4 + t^4$? ✅ 2023-03-30 - [ ] $$\omega=\operatorname{Res}_{X_4}^{\mathbb{P}^3}\left(\frac{d(x / t) \wedge d(y / t) \wedge d(z / t)}{(x / t)^4+(y / t)^4+(z / t)^4+1}\right) .$$ - [x] What is $\Aut(F_4)$? ✅ 2023-03-30 - [ ] $\Aut(F_4) = C_4^3\semidirect S_4$, induced by multiplication by $\zeta_4$ and permuting the coordinates. - [x] What is the number of fixed points of a symplectic automorphism of a K3 of order $n$? ✅ 2023-03-30 - [ ] Note $2 \leq n \leq 8$. - [ ] $$|\operatorname{Fix}(g)|=\frac{24}{n \cdot \Pi_{p \mid n}\left(1+\frac{1}{p}\right)}=: \mu(n) .$$ ## Moduli - [x] What is the primary use of recognizable divisors? ✅ 2023-03-30 - [ ] [AE21 Thm 1.2]: If $\mcd$ is recognizable then the normalization of the log KSBA compactification ,$\qty{ \overline{F_{2d}}^{\mcd}}^\nu$, is semitoroidal for some semifan $\Sigma$ . - [x] What is a monodromy vector? ✅ 2023-03-30 - [ ] For $e\in \Lambda_{2d}$, any $\lambda \in e^\perp/e$ with $\lambda^2 \geq 0$. - [ ] ## Enriques Surfaces - [x] Discuss properties of Enriques surfaces. ✅ 2023-03-30 - [ ] Always algebraic - [ ] Always Kahler - [ ] Always quotients of a K3 by an involution. - [ ] Always elliptic - [x] What is the lattice structure of $H^2(X; \ZZ)$ for an Enriques surface? ✅ 2023-03-30 - [ ] $\rm{II}_{1, 9} \perp \gens{2}$. - [x] What are the plurigenera of an Enriques surface? ✅ 2023-03-30 - [ ] $\sum_n P_n t^n = \sum_{k} t^{2k}$, so 1 iff even and 0 iff odd. - [x] What is $\pi_1 X$ for an Enriques surface? ✅ 2023-03-30 - [ ] $\pi_1 X = C_2$. - [ ] What is a supersingular Enriques surface? - [x] Give an example in equations of an Enriques surface. ✅ 2023-03-30 - [ ] Take $F_4 = V(w^4 + x^4 + y^4 + z^4)$ the Fermat quartic and $T(w, x,y,z) \da (w, ix, -y, iz)$. Then $T^2$ is an involution with 8 fixed points. - [ ] Take $S \da \Bl_8(F_4)/T^2$ which is a K3 with an involution $T$, so $X\da S/T$ is Enriques. - [ ] Or take $X = \widetilde{F_4/T}$, resolving the 8 singular points. - [x] What is the lattice associated to an Enriques surface regarded as a pair $(X, \iota)$ with $X$ a K3? ✅ 2023-03-30 - [ ] $U(2) \oplus E_8(2)$. - [x] What is the lattice structure on $\Num(X)$ for $X$ Enriques? ✅ 2023-03-30 - [ ] $\sgn \Lambda = (1, 9)$ and $\Lambda = H \perp E_8(-1)$. - [x] What is the Enriques lattice? ✅ 2023-03-30 - [ ] $\EE_{10} \da H \perp E_8(-1)$. - [x] Discuss moduli of unpolarized Enriques surfaces. ✅ 2023-03-30 - [ ] Over $\FF_q$ for $p\neq 2$, an irreducible smooth Artin stack of dimension 10. - [ ] Over $\CC$, coarse space of type IV. - [ ] What is the Horikawa model of an Enriques surface? - [ ] What is $W(\EE_{10})$? - [x] What is the level 2 congruence subgroup of $W(\EE_{10})$? ✅ 2023-03-30 - [ ] $\ker \qty{ W(\EE_{10}) \to \Orth_{10}^+(\FF_2) }$. - [x] What is the Reye lattice of an Enriques surface? ✅ 2023-03-30 - [ ] $R_X \da \ts{x\in \Num(X) \st x.R \equiv 0\mod 2 \, \forall \text{ (-2) curves } R}$. - [x] What is the discriminant group of $R_X$? ✅ 2023-03-30 - [ ] $R_X\dual/R_X$. - [x] What is a semi-symplectic automorphism of an Enriques surface? ✅ 2023-03-30 - [ ] For $g\in \Aut(S)$, lift to $\tilde g\in \Aut(\tilde S)$ a K3, then $g$ is semi-symplectic iff either $\tilde g$ is symplectic **or** anti-symplectic. - [ ]