Schedule for: 23w5065 - Around Symmetries of K3 Surfaces

A bidouble cover is a Galois cover whose Galois group is $(\mathbb{Z}/2\mathbb{Z})^2$. If $X\rightarrow Y$ is a bidouble cover there are 3 intermediate covers $Y_i$, $i=1,2,3$ which are $\mathbb{Z}/2\mathbb{Z}$-quotient of $X$. We will consider bidouble covers $X$ of a rational surface $Y$ and we will assume that the surfaces $Y_i$ are either surfaces with $h^{2,0}=0$ or K3 surfaces. Under this condition the transcendental Hodge structure of $X$ splits into the direct sum of sub-Hodge structures of K3-type, each of them geometrically related with a K3 surface. This gives a strong control on the Hodge structure of $X$ and allows one to discuss some of the classic problems related with the Hodge structures of the second cohomology group of surfaces: we first discuss conditions under which $X$ satisfies the Mumford Tate conjecture and the Tate conjecture or enjoys the infinitesimal Torelli property. Then, we provide a classification of the smooth bidouble covers of a minimal rational surface satisfying the previous conditions and we discuss how to modify and generalize the previous construction by considering either singular bidouble covers or the so-called iterated bidouble covers. The talk is based on a joint work with Matteo Penegini.

I will talk about a joint work with L. Giovenzana, A. Grossi and D.C. Veniani in which we prove that the only finite order symplectic automorphism on an O'Grady tenfold is the identity. After having recalled the setting and the main definitions, I will give an idea of the proof focusing on the geometric and lattice-theoretic aspects of it.

We discuss recent progress on the variation of the Mordell-Weil rank in families of elliptic curves over number fields. In the case of elliptic K3 surfaces, we show, under certain conditions, that the set of fibres for which the Mordell-Weil rank is strictly larger than the generic rank is not thin, as a subset of the base of the fibration. This is based on joint work with Hector Pasten.

Inose surfaces (associated with Kummer surfaces of Jacobians of smooth genus-two curves) provide a purely geometric interpretation for a certain duality in string theory. Building on this foundation, I will explain how algebraic K3 surfaces obtained from abelian varieties provide a fascinating arena for string compactification and string dualities as they are not-trivial spaces, but are sufficiently simple to analyze most of their properties in detail. I will then describe recent results where we used families of lattice polarized K3 surfaces of Picard rank 10, 14, and 16 to provide a geometric interpretation, called geometric two-isogeny, for the so-called F-theory/heterotic string duality in eight dimensions with up to 4 Wilson lines. This is joint work with A. Clingher.

In the framework of the compactification of the moduli spaces of prime order non-symplectic automorphisms of irreducible holomorphic symplectic manifolds, a key question is to understand the geometry of limit automorphisms. Starting from a nodal degeneration of cubic threefolds, the general member of the family of Fano varieties of lines of the triple covering branched over the cubic is an IHS manifold equipped with the automorphism induced by the covering. It degenerates to a variety whose singular locus is a K3 surface. I will present recent results obtained in collaboration with Chiara Camere and Alessandra Sarti that explain how the geometry of this K3 surface permits to define a limit automorphism in a suitable moduli space parametrizing pairs of IHS manifolds with automorphism.

In this talk we will study IHS manifolds of $K3^[2]$-type with a symplectic action of $Z_3^4 : A_6$, the symplectic group with the biggest order, and such that they also admit a non-symplectic automorphism. We will show that there are three IHS manifolds that satisfies this, with two of them admitting two possible actions, and particularly we will show that there is a unique IHS manifold of $K3^[2]$-type with finite automorphism group of order 174960, the biggest possible order for the automorphism group of a IHS manifold of $K3^[2]$-type. This is a joint work with Paola Comparin and Romain Demelle.

In this talk, I will first recall the construction of Lagrangian fibrations by Prym varieties starting from a K3 surface with a non-symplectic involution. Then I will discuss a criterion to ensure that the normalizazion of such a fibration is an irreducible symplectic variety. This is joint work in progress with E. Brakkee, A. Grossi, L. Pertusi, G. Saccà and A. Viktorova.

Given a (smooth) projective (complex) surface S and a complete linear (or algebraic) system of curves on S, one defines the Severi varieties to be the (possibly empty) subvarieties parametrizing nodal curves in the linear system, for any prescribed number of nodes. These were originally studied by Severi in the case of the projective plane. Afterwards, Severi varieties on other surfaces have been studied, mostly rational surfaces, K3 surfaces and abelian surfaces, often in connection with enumerative formulas computing their degrees. Interesting questions are nonemptiness, dimension, smoothness and irreducibility of Severi varieties. Whereas it is know that a general primitively polarized K3 surface contains nodal curves of every possible geometric genus g, and that the dimension of the corresponding Severi varieties are precisely g, very little has been known on special K3 surfaces, such as for instance the ones admitting an Enriques involution, that is, a fixed point free involution, so that their quotients are Enriques surfaces. In this talk I will present recent results about Severi varieties on Enriques surfaces, obtained with Ciliberto, Dedieu and Galati.

Many moduli spaces can be described as ball quotients. Examples include the Deligne-Mostow varieties, moduli of cubic surfaces and certain moduli spaces of lattice-polarized $K3$ surfaces. Here I will discuss the geometry of some of these examples, including their topology and different (partial) resolutions. I will also comment on the relationship with the Minimal Model Program.

Building on the work of Shioda and Shioda-Usui on "excellent families" of rational elliptic surfaces, we study several related families of rational elliptic surfaces with two additive fibers. In each case the moduli space is a complement of hyperplanes in a hypersurface S, which is also the moduli space of suitably polarized K3 surfaces X obtained by quadratic base change ramified at the additive fibers. We illustrate this with the example of additive fibers of types II and IV, where S is the quintic fourfold invariant under the Weil group of E_6, and X is a quartic surface with a 40_{12} configuration of lines. Other examples includes cases where S is the Segre cubic or Igusa quartic threefold, the self-dual sextic fourfold with W(D_6) automorphism, and new moduli spaces of dimension 5 and 6.

In a paper of 2019 Boissière–Camere–Sarti prove that there exists an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkähler fourfolds of K3[2]-type with a non-symplectic automorphism of order three, whose invariant lattice is generated by a class of square 6. Then, the authors study the degeneration of the automorphism along a generic nodal hyperplane, proving the existence of a birational map from the locus of nodal cubic threefolds to some moduli space of hyperkähler fourfolds of K3[2]-type with a non-symplectic automorphism of order three belonging to a different family. In the exceptional locus of this morphism there are several interesting cubic threefolds. Therefore, I will present a generalization of their result to some non-generic nodal cases.

We study the hypergeometric functions associated to five one-parameter deformations of K3 quartic hypersurfaces in projective space, each admitting a symplectic group action. We match the Picard–Fuchs differential equations to factors of the zeta function, and we write the result in terms of global L-functions. We obtain a complete, explicit description of the motives for these pencils in terms of hypergeometric motives. This talk describes joint work with Charles Doran, Tyler Kelly, Adriana Salerno, Steven Sperber, and John Voight.

J.W. with A. Sarti. Let $B$ be an abelian surface. Suppose that there exists an order $3$ automorphism $j_A$ acting on $A$ symplectically. The quotient surface $A/J_A$ has nine cuspidal singularities. The minimal resolution $Km(A,j_A)$ of $A/J_A$ is called a generalized Kummer surface; it contains a configuration $\mathcal{C}$ of type $9A_2$, which means nine disjoint pairs of $(-2)$-curves $C,C'$ such that $CC'=1$, which curves are above the singularities. A Kummer structure on a generalized Kummer surface $X$ is an isomorphism class of pairs $(B,j_B)$ where $B$ is an abelian surface and $j_B$ an order $3$ automorphism such that the associated generalized Kummer surface $Km(B,j_B)$ is isomorphic to $X$. One wants to understand these generalized Kummer structures. Thanks to results of Barth, the data of a Kummer structure is equivalent to the orbit under the automorphism group of $A$ of nine disjoint $A_2$-configurations. Using the Pell-Fermat equation and some geometric results, we constructy new $9A_2$-configurations $\mathcal{C}'$ on a given generalised Kummer surface. Using the Torelli theorem, one obtains the following alternative: a) Either there exists no automorphism sending $\mathcal{C}$ to $\mathcal{C}'$, then we get a non-trivial Kummer structure, b) Or there exists an automorphism sending $\mathcal{C}$ to $\mathcal{C}'$, and then we can describe the action of such automorphism on the N\'eron-Severi lattice and obtain in that way non-trivial elements of the automorphism group of the generalized Kummer surface.

Given a smooth quartic K3 surface $S \subset P^3$, Gizatullin was interested in which automorphisms of $S$ are induced by Cremona transformations of $P^3$. Later on, Oguiso answered it for some interesting examples and he posed the following natural question: Is every automorphism of finite order of any smooth quartic surface $S \subset P^3$ induced by a Cremona transformation? In this talk, we will give a negative answer to this question by constructing a family of smooth quartic K3 surfaces $S_n$ with Picard number two such that $Aut(S_n) = D_\infty$ together with an involution of $S_n$ that is not derived by any element of $Bir(P^3)$. More precisely, we will prove that no element of $Aut(S_n)$ is induced by an element of $Bir(P^3)$.

Several projective constructions of families of holomorphic symplectic varieties are known but they are all of K3 type. I will describe work in progress towards constructing an explicit family of holomorphic symplectic 4-folds of generalised Kummer type.

Finite symplectic automorphism groups on complex K3 surfaces have famously been classified by Mukai, based on work of Nikulin. I will report on joint work with Hisanori Ohashi which aims to extend this to positive characteristic. While the tame case retains a close connection to the Mathieu group M_{23} (extending work of Dolgachev-Keum), we will develop a unified approach using symmetries of the Leech lattice which also covers all wild cases.

Having an automorphism is a non-trivial property for a complex K3 surface. Indeed, if $X$ is a generic complex K3 surface of degree $d\geq 4$, then the only automorphism of $X$ is the identity. If $X$ is a generic of degree $d=2$, then $X$ admits only one involution beside the identity map. Hence a natural question arises: given a fixed positive even integer $d$, how special is it for a K3 surface of degree $d$ to admit an involution? More precisely, what is the minimal Picard number $h_d$ for a K3 surface of degree $d$ in order to admit an involution as automorphism? In this talk we are going to show that $h_d=2$ for every $d\geq 4$. This is joint work with Wim Nijgh and Daniel Platt.

Landau-Ginzburg (LG) models consist of the data of a quotient stack X and a regular complex-valued function W on X. Here, geometry is encapsulated in the singularity theory of W. One can find that LG models are deformations of many Calabi-Yau varieties in some sense. For example, if the Calabi-Yau is a hypersurface in a smooth projective variety Z cut out by a polynomial f, then one can take X to be the canonical bundle of Z with function W=uf, where u is the bundle coordinate—when the hypersurface is smooth, the critical locus of uf will indeed just be the hypersurface. Exoflops were introduced by Aspinwall as a way to effectively find new birational models of the quotient stack to get new geometries. They effectively create new GIT problems of partial compactifications of X, expanding the tractable birational geometries related to Z. We will explain this technique, provide some foundational results about this, and then provide some new applications proven recently for Calabi-Yau varieties with nontrivial scaling symmetry groups. This talk contains joint work with D. Favero (UMinn), C. Doran (Bard), A. Malter (Birmingham).

The problem of finding a presentation for the Cox ring of a Mori dream space (Cox ring is finitely generated) is interesting and difficult. For K3 surfaces, it is known that the Cox ring is finitely generated if and only if its effective cone is polyhedral, or equivalent if its auto- morphism group is finite. K3 surfaces with this last property have been classified in a series of classical works and for those with Picard number ≥ 3 there are a finite number of families. In this talk we will see a general theorem about Cox rings of K3 surfaces (not necessarily Mori dream) and some examples where we have applied this result together with other techniques based on exact sequences of the Koszul type, allowing us to obtain the degrees of a set of generators of the Cox ring of K3 Mori dream surfaces. This work is in collaboration with M. Artebani, A. Laface and X. Roulleau.

Mukai classified all symplectic groups of automorphisms of K3 surfaces as possible subgroups of one of the Mathieu groups. Since then, the proof of Mukai's theorem has been simplified using lattice theoretical techniques, and extended to higher dimensional hyperkähler manifolds. In two joint works with L. Giovenzana, A. Grossi and C. Onorati, we studied possible cohomological actions of symplectic automorphisms of finite order on the two sporadic deformation types found by O'Grady in dimension 6 and 10. In particular, we showed that all symplectic automorphisms in dimension 10 are trivial. In my talk, I will explain the connection between our proof and the sphere packing problem.