--- created: 2023-03-26T11:58 updated: 2024-06-23T16:28 aliases: - Enriques surface - Enriques - Enriques surfaces links: "[[Flashcards Surfaces|Flashcards Enriques Surfaces]]" --- See [K3](K3%20surfaces.md). # Facts ![](2024-06-23-7.png) ![](2024-01-24-6.png) ![](2024-06-23-8.png) ![](2024-06-23-10.png) [@cossec1989enriques-surfaces] - Hodge diamond: ![](2024-01-24-2.png) - $q(S) = 0 = h^1(\OO_S)$. - $p_g(S) = 0 = h^0(K_S) = h^2(\OO_S)$ - $p_a(S) = ?$ - $\Pic(S) = ?$ - $\chi(\OO_S) = 1$ - $\rho(S) = 10$ - If $E$ is a rank 2 vector bundle on $S$, $\chi(E) = 2 + {1\over 2} c_1^2(E) - c_2(E)$. - $K_S = F_A - F_B = F_B - F_A$ where $F_A, F_B$ are the canonical half-pencils. - Always elliptic or quasi-elliptic. - Moduli lattice: $L = U(2) \oplus E_8(2) \da E_{10}(2)$. - $\chi(\K3, \OO_{\K3}) = 1 + (1 + 20 + 1) + 1 = 24 = \dim_\CC H^*(X; \CC)$. # Motivation [@cossec1989enriques-surfaces] ![](2024-01-24-3.png) - Castelnuovo: $q(X) = p_g(X) = P_2(X) = 0 \implies X$ is rational. - Question: is it true that $q(X) = p_g(X) = 0$ is sufficient? So $P_2(X) \neq 0$, but $X$ is still necessarily rational? - Answer: no, viz. Enriques surfaces $S$ with $q(S) = p_g(S) = 0$ but $P_{2n} = 1$ for all $n\geq 0$ which are not rational. Also $\kappa(S) = 0$. - Birational model of this: the Enriques sextic, a non-normal degree 6 surface in $\PP^3$ passing through the edges of the coordinate tetrahedron with multiplicity 2. ![](2024-01-11.png) - An Enriques surface $S$ is called **unnodal** if $\mathcal{R}(S)=\emptyset$, that is, if $S$ does not contain any $(-2)$-curves. Otherwise, $S$ is called **nodal**. [@cossec1989enriques-surfaces, Def. 2.3.1.] - Every Enriques surface carries at least one genus one pencil. [@cossec1989enriques-surfaces, Cor. 2.3.4.] - # Enriques Surfaces Definitions - An Enriques surface $Y$ is a (projective) complex surface with universal double cover a [K3 surface](K3%20surfaces.md). One knows that $H^2(Y, \mathbb{Z})=\mathbb{Z}^{10} \oplus \mathbb{Z}_2$ and that the cup-product provides $H^2(Y, \mathbb{Z}) /$ torsion $=\mathbb{Z}^{10}$ with the structure of a [lattice](lattice.md) $M$ of signature $(1,9)$. [(Barth, W., Peters, C., 1982. Automorphisms of Enriques Surfaces.)](zotero://select/library/items/Q9AY2RNU) - Every Enriques surface is an elliptic fibration over $\mathbb{P}^1$ with two multiple fibres $2 F_A$ and $2 F_B . F_A$ and $F_B$ are reduced curves and they are called half-pencils. ![](attachments/Pasted%20image%2020220323230047.png) ![](attachments/Pasted%20image%2020220515013021.png) ![](attachments/2023-03-04enriques.png) # Types of Enriques surfaces ![](2024-01-24-7.png) # Enriques lattice ![](2023-04-05-16.png) # Current research Simon Brandhorst: Computation of the automorphism group of Enriques surfaces > The Morrison–Kawamata cone conjecture predicts that the action of the automorphism group of a Calabi-Yau variety on its effective nef cone admits a fundamental domain which is a rational polyhedral cone. > > The conjecture is wide open in general. But Namikawa has verified it for Enriques surfaces. It follows that an Enriques surface admits up to the action of the automorphism group only finitely many smooth rational curves, elliptic fibrations, projective models of a given degree and its automorphism group is finitely generated and in fact finitely presented. > > Naturally, enumerative questions arise: > > - Can one explicitly describe a fundamental domain? > > - How many smooth rational curves or elliptic fibrations are there up to the action of the automorphism group? > > - Can one give generators for the automorphism group? Andreas Leopold Knutsen: Moduli spaces of polarized Enriques surfaces and their possible unirationality > Moduli spaces of polarized Enriques surfaces (parametrizing  isomorphism classes of pairs (S,H), where S is a smooth compact, complex Enriques surface, and H is an ample line bundle on S) have several components, even if one fixes the degree of the polarization. I will > > present some new results showing how one can determine the various irreducible components in terms of intersection with suitable sequences of isotropic divisors, and how one can prove that infinitely many of these components are unirational. > > In particular, there is a particular component dominating all others, and according to an announced (but yet unpublished) result of Gritsenko, this space should be of general type, showing that not all moduli spaces can be of negative Kodaira-dimension. > > See [Irreducible unirational and uniruled components of moduli spaces of polarized Enriques surfaces](https://arxiv.org/abs/1809.10569) Yulieth Prieto-Montañez: K3 Surfaces and an Adversary of Enriques Surfaces > It is a well-established fact that complex Enriques surfaces can be represented as quotients of K3 surfaces through an involution group without fixed points. Drawing inspiration from this geometrical construction, which arises from considering cyclic coverings, we delve into the study of K3 surfaces as double covers of rational surfaces. > > In this talk, the involutions under consideration are those that fix curves, differing from the Enriques construction. We explore elliptic fibrations of K3 surfaces induced by conic bundles of Del Pezzo surfaces. Our findings reveal the possibility of obtaining non-trivial elliptic fibrations on the K3 side. Following Kodaira's classification of singular fibers, we classify all elliptic fibrations on the K3 surfaces admitting these involutions. This classification can be derived through classes of conic bundles on the rational surfaces that align with the geometrical model, in accordance with the findings by Manin and Dolgachev. Davide Cesare Veniani: Non-degeneracy of Enriques surfaces > Enriques' original construction of Enriques surfaces dates back to 1896. It involves a 10-dimensional family of sextic surfaces in the projective space which are non-normal along the edges of a tetrahedron. An even earlier construction of Enriques surfaces is due to Reye and is known as Reye congruences.  > > In a series of joint works with G. Martin and G. Mezzedimi, we have now settled two questions: (1) Do all Enriques surfaces arise through Enriques' construction? (2) Do all nodal Enriques surface arise as Reye congruences? Franco Giovenzana, Annalisa Grossi (Université Paris-Saclay): Enriques manifolds > Enriques manifolds were introduced by Boissière-Nieper-Wisskirchen-Sarti and Oguiso-Schröer as a generalisation in higher dimension of Enriques surfaces. In this talk we will revise the basics and the known examples of Enriques manifolds, then we will discuss a possible way to get new examples.