# Young Perspectives in Algebraic Geometry

"Young Perspectives in Algebraic Geometry" (YPAG) is an international conference which will take place from 7-9 March 2018 at the Mathematisches Institute, Universität Bayreuth.

The main purpose of YPAG is to gather people from Germany and across Europe to present and discuss the research that young algebraic geometers are currently undertaking.**Confirmed speakers:**

Maria Donten-Bury (University of Warsaw)

Víctor González-Alonso (Leibniz Universität Hannover)

Liana Heuberger (University of Warwick)

Andreas Hochenegger (Università Milano)

Grzegorz Kapustka (Jagiellonian University Cracow)

Roberto Laface (TU München)

Diletta Martinelli (University of Edinburgh)

Fabio Tanturri (Université de Lille)

Luca Tasin (Universität Bonn)

Alan Thompson (University of Cambridge)

## ABSTRACTS

**Maria Donten-Bury** Cox rings and crepant resolutions of 3-dimensional quotient singularities

Abstract: In this talk I will show how Cox rings can be used to study resolutions of quotient singularities. In particular, I will present results of a joint project with Maksymilian Grab, which concerns crepant resolutions of quotient singularities in dimension 3. In certain cases we apply Cox rings to obtain information on geometry of resolutions via Geometric Invariant Theory, we determine the number of all crepant resolutions of considered singularities and investigate the combinatorial picture of birational maps between them.

**Víctor González-Alonso** On the unitary rank of fibred surfaces.

Abstract: The unitary rank of a fibred surface is defined as the rank of the flat unitary summand in the so-called second Fujita of the Hodge bundle of the fibration (the direct image of the dualizing sheaf). Understanding this flat summand, and in particular its rank, amounts to understand the locally constant part of the Variation of Hodge Structures associated to the fibration. In this talk I will present an upper bound on the unitary rank, depending on the genus and the Clifford index of a general fibre. Actually, it is the same bound we obtained in a previous work with Barja and Naranjo for the relative irregularity, although the unitary rank can very well be much bigger than the relative irregularity. This is joint work with Lidia Stoppino and Sara Torelli.

**Liana Heuberger** Title: Fano fourfolds with small invariants

Abstract: One of the more concrete ways of approaching mirror symmetry is to study toric degenerations of (almost) smooth Fano varieties. The polytopes associated to these toric varieties are usually reflexive, as for surfaces and threefolds they are enough to recover the famous classification of Mori and Mukai. However, this doesn't hold in dimension four: we miss all Fano fourfolds with ${p}_{1}(-{K}_{X})<6$ if we only consider smooth varieties given by a reflexive polytope. I will explain an attempt at finding the rest.

**Andreas Hochenegger **Formality of $\mathbb{P}$-objects.

Abstract: An Calabi-Yau-object in a $k$-linear triangulated category is called a $\mathbb{P}$-object, if its derived endomorphism ring is isomorphic to $k[t]/{t}^{n}$. They were first studied by Daniel Huybrechts and Richard Thomas as generalisations of spherical objects. Similar to the spherical case, $\mathbb{P}$-objects induce autoequivalences which are called $\mathbb{P}$-twists. Recently, Ed Segal showed how an arbitrary autoequivalence can be written as a spherical functor. For a $\mathbb{P}$-twist, he needs the assumption that the endomorphism ring of the $\mathbb{P}$-object is formal. In this talk, I will introduce the concept of formality and present a proof of the formality of configurations of $\mathbb{P}$-objects. This is based on a joint work with Andreas Krug.

**Grzegorz Kapustka **On the Morin problem

Abstract: We will study the Morin problem and present a method of classification of finite complete families of incident planes in ${\mathbb{P}}^{5}$. As a result we prove that there is exactly one, up to Aut(${\mathbb{P}}^{5}$), configuration of maximal cardinality 20 and a unique one parameter family containing all the configurations of 19 planes. The method is to study projective models of appropriated moduli spaces of twisted sheaves on K3 surfaces. This is a joint work with A. Verra.

**Roberto Laface Picard** numbers of abelian varieties in all characteristic

Abstract: I will talk about the Picard numbers of abelian varieties over $\mathbb{C}$ (joint work with Klaus Hulek) and over field of positive characteristic. After providing an algorithm for computing the Picard number, we show that the set ${R}_{g}$ of Picard numbers of $g$-dimensional abelian varieties is not complete if $g\ge 2$, that is there are gaps in the sequence of Picard numbers seen as a sequence of integers. We will also study which Picard numbers can or cannot occur, and we will deduce structure results for abelian varieties with large Picard number. In characteristic zero we are able to give a complete and satisfactory description of the overall picture, while in positive characteristic there are several pathologies and open questions yet to be answered.

**Diletta Martinelli** On the geometry of contractions of the Moduli Space of sheaves of a K3 surface

Abstract: I will describe how recent advances have made it possible to study the birational geometry of hyperkaehler varieties of K3-type using the machinery of wall-crossing and stability conditions on derived categories as developed by Tom Bridgeland. In particular Bayer and Macrì relate birational transformations of the moduli space M of sheaves on a K3 surface X to wall-crossing in the space of Bridgeland stability conditions Stab(X). I will explain how it is possible to refine their analysis to give a precise description of the geometry of the exceptional locus of any birational contractions of M.

**Fabio Tanturri** Orbital degeneracy loci

Abstract: In a joint project with Vladimiro Benedetti, Sara Angela Filippini, and Laurent Manivel, we introduce a new class of varieties, called orbital degeneracy loci. They are modelled on any orbit closure in a representation of an algebraic group and generalize classical degeneracy loci of morphisms between vector bundles or zero loci of sections. In this talk I will introduce some tools to understand and study this new class of objects; with such techniques, we can exploit our construction to produce several interesting examples of projective varieties, in particular varieties with trivial or negative canonical bundle.

**Luca Tasin **On a conjecture of Kawamata for weighted complete intersections.

Abstract: Let X be a smooth (or mildly singular) projective variety and let H be an ample line bundle on X. Kawamata conjectured that if H-K_X is ample, then the linear system |H| is not empty. I will explain that the conjecture holds true for weighted complete intersections which are Fano or Calabi-Yau, relating it with the Frobenius coin problem, a classical arithmetic question. This is based on a joint work with M. Pizzato and T. Sano.

**Alan Thompson** ADE Surfaces

Abstract: In 2000, Losev and Manin showed that a certain moduli space of points on the projective line can be identified with a toric variety associated to a root lattice of type A. I will show that Losev and Manin's moduli space may be reinterpreted as a moduli space of rational surfaces with involution and, moreover, that variations on this construction produce surfaces whose moduli spaces are toric varieties associated to root lattices of types D and E. Finally, if time allows, I will show that there exist modular families on these moduli spaces, which extend to modular families of stable surfaces on a natural compactification. The explicit descriptions of these modular families show surprising links to the representation theory of Lie algebras. This is joint work with Valery Alexeev.