![]() ![]() We construct four-dimensional families of categories that are deformations of D^b(A) over an algebraic space. I present a construction that addresses both issues. Polarised GKVs have four-dimensional moduli spaces, yet arise from moduli spaces of stable sheaves on abelian surfaces only over three-dimensional subvarities. Generalised Kummer varieties (GKV) are Hyperkaehler varieties arising from moduli spaces of stable sheaves on abelian surfaces. In contrast, the derived category of an abelian surface A has a six-dimensional space of deformations moreover, based on general principles, one should expect to get "algebraic families" of their categories over four-dimensional bases. Title: Non-commutative abelian surfaces and generalised Kummer varietiesĪbstract: Polarised abelian surfaces vary in three-dimensional families. This is joint work with Hans-Christian von Bothmer, Brendan Hassett and Yuri Tschinkel. Time permitting, I will also discuss some results about G-Del Pezzo surfaces and some classes of higher-dimensional G-Fano varieties. I will exhibit examples of nonlinearisable but stably linearisable actions of finite groups on smooth cubic fourfolds that show that the natural equivariant analogs of existing rationality conjectures for cubic fourfolds do not hold. Title: Equivariant birational geometry of cubic fourfolds and derived categoriesĪbstract: In the talk I will discuss several aspects of equivariant birationality from the perspective of derived categories. If time permits, we will discuss the significance of our formula and potential applications. ![]() The calculation involves constructing a resolution by means of subsequent blow-ups. Then we can compare the two universal Brill-Noether classes on the two compactified Jacobians by pulling one of them back along the (rational) identity map. More precisely, fix two stability conditions for universal compactified Jacobians that are on different sides of a wall in the stability space. Title: A wall-crossing formula for universal Brill-Noether classesĪbstract: We will discuss an explicit graph formula, in terms of boundary strata classes, for the wall-crossing of universal (=over the moduli space of stable curves) Brill-Noether classes. This is joint work with Benjamin Nill and Hendrik Süss. I will then present a theorem that on toric surfaces and toric varieties of Picard rank 2 there exists an ample line bundle such that the tangent bundle is stable if and only if the underlying variety is an iterated blow up of projective space. Title: Stability of Toric Tangent bundlesĪbstract: In this talk I will give a brief introduction to slope stability and present a combinatorial criterion for the tangent bundle on a polarised toric variety to be stable in terms of the lattice polytope corresponding to the polarisation. MS.04 - note unusual room! Speaker: Hal Schenck (Auburn University) Title: On rationality in families and equivariant birational geometryĪbstract: In this talk I will recall some developments, connected with rationality in families of varieties, and explain some analogous developments in equivariant birational geometry, obtained in recent joint work with Brendan Hassett and Yuri Tschinkel. Speaker: Andrew Kresch (University of Zurich) It can also be seen as the midpoint of the standard monoidal transformation PP^3 -> PP^3 given by (x,y,z,t) |-> (1/x,1/y,1/z,1/t). This construction leads naturally to the Enriques-Fano variety that is the toric variety (PP^1 x PP^1 x PP^1)/(☑) embedded in PP^13 by monomials corresponding to the face-centred cube. Putting both of these linear systems together into a graded ring gives a toric extraction of the 6 coordinate lines. ![]() An Enriques sextic is a sextic surface that passes doubly through the 6 coordinate lines. It automatically passes through the 6 coordinate lines. Title: Cayley cubics and Enriques sexticsĪbstract: A Cayley cubic is a cubic surface of PP^3 with nodes at the 4 coordinate points. The algebraic geometry seminar in Term 3 2022/2023 will usually meet on Wednesdays at 3pm in MS.03, though we may sometimes change to allow speakers from other time zones. ![]()
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