Pieter Belmans (University of Antwerp): Quivers of exceptional collections on smooth projective varietiesWe study the finite-dimensional algebras arising as endomorphism algebras of exceptional collections on smooth projective varieties. We survey some basic and less basic results and provide a couple of new ones. Since this is a rather broad question we will be particularly interested in the extreme case: hereditary algebras and smooth projective surfaces. This is joint work with Theo Raedschelders. Pawel Borowka (Jagiellonian University Krakow): Non-simple abelian varietiesThe Poincare's Reducibility Theorem states that for a non-simple abelian variety and its abelian subvariety, one can find a complementary abelian subvariety and an isogeny from the product of subvarieties. In the moduli space of principally polarised abelian varieties, we discuss the irreducible components of the locus of non-simple abelian varieties as those with fixed dimension and type of its abelian subvariety. We also present equations on non-simple principally polarised abelian varieties in the Siegel space. Furthermore, we give some applications of the above result to Jacobians involved in Prym constructions. Matthew Dawes (University of Bath): Irreducible Symplectic Manifolds and Orthogonal Modular formsI shall give an overview of how geometric questions about the moduli of Irreducible symplectic manifolds can be answered by studying modular forms for the orthogonal group. Ruadhai Dervan (University of Cambridge): Alpha invariants and K-stability on Fano varieties To a polarised variety (X,L) one can associate an invariant called the alpha invariant, which measures singularities of divisors in the linear system associated to L. A theorem of Odaka-Sano says that in the Fano case a certain lower bound on the alpha invariant implies that the manifold (X,-K_X) is K-stable (which is the algebraic counterpart to a famous theorem of Tian concerning Kahler-Einstein metrics). This poster describes a a criterion in terms of the alpha invariant for certain other polarisations to be K-stable, together with examples. Carmelo Di Natale (University of Cambridge): A Period Map for Global Derived StacksIn the Sixties Griffiths constructed a holomorphic map, known as the local period map, which relates the classification of smooth projective varieties to the associated Hodge structures. Fiorenza and Manetti have recently described it in terms of Schlessinger's deformation functors and, together with Martinengo, have started to look at it in the context of Derived Deformation Theory. In this poster we propose a rigorous way to lift such an extended version of Griffiths period map to a morphism of derived deformation functors and use this to construct a period morphism for global derived stacks. Tom Ducat (University of Warwick): Unprojection and 3-Fold Divisorial ExtractionsUnprojection is a useful tool for explicitly constructing algebraic varieties in low codimension. This poster will describe some recent work in constructing 3-fold divisorial extractions that contract a divisor to a singular curve. Andrea Fanelli (Imperial College London): On the fibres of Mori fibre spacesWe are interested in understanding when a given Fano variety can be realised as a general fibre of a Mori fibre space. We are able to give two criteria, one sufficient and one necessary, which turn into a characterisation in the rigid case. As an application, we give a complete answer in the case of surfaces, an almost exhaustive answer for smooth threefolds and flag varieties and a further characterisation on the polytope in the toric case. An interesting connection with K-semistability is also investigated. This is a joint work with Giulio Codogni, Roberto Svaldi and Luca Tasin. Simon Hampe (University of Warwick): Tropical moduli spaces of rational weighted stable curvesWeighted stable rational curves have been proposed by Brendan Hassett as a general way to compactify the moduli space of smooth rational n-marked curves. One assigns weights 0 < w_i <= 1 to the marked points and tweaks the stability condition such that boundary strata correspond to nodal curves with "more than 2" special points on each component (each node is counted with multiplicity 1 and each marked point with its assigned weight). This can be translated to tropical curves using the dual graph operation. What we obtain are metric trees with n labeled ends, such that each vertex has valence "greater 2" (again, count interior edges with weight 1 and ends with their assigned weight). In the case where w = (1,...,1,e,...,e) for small e, we show that the tropical space of these curves is in fact the Bergman fan B(M(G)) of a graphic matroid. The corresponding graph G can be defined purely in terms of the weights and the subdivision along combinatorial types is a nested set subdivision. Analogously to what Maclagan and Gibney have shown for M_0,n, we can embed the open part of the algebraic space M(1,..,1,e...e) such that it tropicalizes onto its tropical counterpart. In addition, its closure in the toric variety defined by the tropical fan is again the closed moduli space of weighted stable curves. For general weights, we still obtain the tropical moduli space as a skeleton of the Berkovich analytification of the algebraic space. This is joint work with Renzo Cavalieri, Hannah Markwig, and Dhruv Ranganathan. Thomas Hawes (University of Oxford): Geometric Invariant Theory for Non-Reductive GroupsWe will give an outline of some results for geometric invariant theory (GIT) for non-reductive group actions on projective varieties. When a reductive group G acts on a projective variety X, Mumford showed how to find an open subset X^s of X of "stable points" (depending on a linearisation of the action) that admits an orbit space variety X^s/G, and this quotient admits a canonical compactification X//G. We will describe some conditions on actions of non-reductive groups which yield a similar---though not identical---picture to that for reductive groups, and which extend the work of Brent Doran and Frances Kirwan on actions of certain unipotent groups. We also describe a toy example to demonstrate the theory. Seung-Jo Jung (University of Warwick): Moduli of G-constellationsFor a finite group G in GL(n), a G-equivariant coherent sheaf F on C^n is called a G-constellation if H^0(F) is isomorphic to the regular representation of G. This poster presents a description of the moduli spaces of G-constellations using toric geometry. Martin Kalck (University of Edinburgh): Triangulated categories in singularity theoryIn the first part, we consider two interesting triangulated categories associated with Gorenstein singularities: the singularity category of Buchweitz and Orlov and the relative singularity category of a non-commutative (Auslander) resolution, which was studied in joint work with Igor Burban. In joint work with Dong Yang, we show that these categories mutually determine each other in the case of ADE-singularities in any Krull dimension. Knörrer's periodicity theorem yields a wealth of non-trivial examples. The general dg framework developed in this context inspired a structure theorem for Frobenius categories, which we used to describe the Iyama & Wemyss triangulated category associated with an arbitrary rational surface singularity. Our result may be seen as a generalization of Auslander's algebraic McKay Correspondence. This is joint work with Osamu Iyama, Michael Wemyss & Dong Yang. Nikon Kurnosov (Laboratory of Algebraic geometry and its applications, Higher School of Economics): Inequalities with Betti and Hodge numbers for hyperkaehler manifoldsA hyperkähler manifold is a Riemannian manifold M endowed with three complex structures I, J and K, satisfying quaternionic relations and such that the metric on M is Kähler with respect to these complex structures. These manifolds are simply connected and the space of its global holomorphic two-forms is spanned by a symplectic form. In present work we study Betti numbers of compact irreducible hyperkähler manifolds. Due to works of Guan it is known that possible b_2 numbers for 4-dimensional hyperkähler manifolds are between 3 and 8 and also 23. In case of dimension 6 and more we obtain several inequalities involving Betti and Hodge numbers.Kyoung-Seog Lee (Seoul National University): Derived categories of surfaces isogenous to a higher productLet S=(C \times D)/G be a surface isogenous to a higher product of unmixed type with p_g=q=0. We study exceptional sequences of line bundles on S. The orthogonal complements of the admissible subcategories generated by exceptional sequences of maximal length in the derived category of S are quasiphantom categories. We compute the Hochschild cohomologies of the quasiphantom categories and prove that for some exceptional sequences we made the DG algebras of endomorphisms are deformation invariant. Chunyi Li (Edinburgh University): Minimal model program for deformations of Hilb(P2)We describe the birational geometry of deformations of Hilbert schemes of points on P2. On one hand, we complete the picture in the paper of Arcara, Bertram, Coskun and Huizenga by giving an explicit correspondence between the stable base locus walls on the Neron-Severi space and the actual walls on the Bridgeland stability space. On the other hand, we show that the birational geometry of a deformed Hilb P2 is different from that of Hilb P2. Diletta Martinelli (Imperial College London) The Magic of \bar{F}_pI will present a recent joint work with Jakub Witaszek (University of Bonn) and Yusuke Nakamura (University of Tokyo) on the Base Point Free Theorem for log canonical threefold over the algebraic closure of a finite field. For varieties with this class of singularities, the results is false over any different field. I will explain some of the reasons why \bar{F}_p is so peculiar. Piotr Pokora (Pedagogical University of Cracow): Local Negativity and Containment ProblemsThe aim of my poster is to present recent developments related to the negativity of curves on algebraic surfaces. I present a certain estimation for a lower bound of self-intersection of negative curves on blow-ups of \mathbb{P}^2 in s arbitrary points and then I show the relations between reduced negative curves and the so-called containment problem for ideals of points on \mathbb{P}^2. These results appeared (and will appear) in of two papers: A counterexample to the containment I^{(3)}\subset I^2 over the reals -- arXiv:1310.0904. The second paper is under construction and will appear as a preprint soon on arxiv. Taro Sano (Max Planck Institute, Bonn): Elephants of Fano threefolds In the classification of Fano threefolds, the anticanonical linear system plays an important role. If there is an anticanonical member with only mild singularities, it is useful in the classification. I will present several features around this topic. Mattia Talpo (Max Planck Institute, Bonn): Infinite root stacks of logarithmic schemes and moduli of parabolic sheavesWe introduce the notion of infinite root stack of a logarithmic scheme. This is some kind of algebraic analogue of the "Kato-Nakayama space" of a log analytic space, and its geometry is closely connected to the log geometry of the log scheme. In particular, we apply this construction to obtain a moduli theory for parabolic sheaves with arbitrary rational weights on a log scheme (under certain hypotheses). Rebecca Tramel (University of Edinburgh): Stability on surfacesWe consider Bridgeland stability on surfaces with a curve of negative self-intersection, and look at wall-crossing. Anna Wißdorf (Freie Universität Berlin): Complementary Polyhedron of Higgs BundlesGiven a vector bundle on a curve, there is a unique Harder-Narasimhan filtration that codifies the (in-)stability of the vector bundle. K. Behrend generalised this result to group schemes on curves by constructing a so-called complementary polyhedron. We will construct a complementary polyhedron for Higgs bundles on curves, which also implies the existence and uniqueness of a Harder-Narasimhan filtration for Higgs bundles. |

1st BrAG meeting >