KP solutions from nodal curves, and the Schottky problem (slides)

Claudia Fevola

The study of solutions to the The Kadomtsev-Petviashvili (KP) equation yields interesting connections between integrable systems and algebraic curves. In this talk, I will discuss solutions to the KP equation whose underlying algebraic curves undergo tropical degenerations. In these cases, the Riemann’s theta function becomes a finite exponential sum supported on a Delaunay polytope. I will introduce the Hirota variety which parametrizes KP solutions arising from such a sum. I will then focus on a special case: the Hirota variety associated to a rational nodal curve. Of particular interest is an irreducible subvariety that is the image of a parameterization map. Proving that this is a component of the Hirota variety entails solving a weak Schottky problem for rational nodal curves. This talk is based on joint works with Daniele Agostini, Yelena Mandelshtam, and Bernd Sturmfels. 

How many cubic surfaces are tangent to 19 general lines? (slides)

Andreas Kretschmer

Given an N-dimensional family F of subvarieties of projective n-space, the number of members of F tangent to N general linear spaces is called a characteristic number for F. More general contact problems can very often be reduced to the computation of these characteristic numbers. However, determining the characteristic numbers themselves can be very difficult, and very little is known even for the family of smooth degree d hypersurfaces of projective n-space as soon as n,d>2. In this talk I will survey the construction of a so-called 1-complete variety of cubic hypersurfaces, generalizing Aluffi's variety of complete plane cubic curves to higher dimensions. The main goal is to answer the question posed in the title and more of a similar flavor, and to give an intuitive geometric meaning to the "exceptional points" of our 1-complete variety of cubic hypersurfaces. 

Tautological systems for projective homogeneous spaces 

Paul Görlach

Orbit closures of algebraic group representations include many classical algebraic varieties. Some of their geometry reflects in specific systems of differential equations. They can be studied as modules over the Weyl algebra and in good situations carry the rich structure of a mixed Hodge module. In this talk, we first revisit the case of representations of algebraic tori which corresponds to toric varieties and their associated GKZ-hypergeometric systems. Then we broaden the scope to the general situation and discuss tautological systems from a functorial standpoint. Finally, we turn our attention to the case of transitive actions on projective varieties to obtain explicit (non-)vanishing results and describe rank formulas in terms of geometry. This talk is based on joint work with Thomas Reichelt, Christian Sevenheck, Avi Steiner and Uli Walther. 

Minimal rational curves on complete symmetric varieties            (joint work with Shinyong Kim and Nicolas Perrin) 

Michel Brion

A family of rational curves on a projective variety X is called minimal if the subfamily of curves through a general point of X is non-empty and proper. The minimal families and the associated varieties of minimal rational tangents (consisting of the tangent directions of the curves through a given point) yield important geometric invariants of X. The talk will discuss the problem of describing the minimal rational curves when X has an action of an algebraic group with an open orbit, and its recent solution for complete symmetric varieties (where the open orbit is a symmetric space). 

C* action and birational geometry

Jarosław Wiśniewski

I will review recent results linking C* action and related quotients with birational geometry of projective varieties. The talk will be based on joint work with Gianluca Occhetta, Eleonora Romano, and Luis Sola Conde.

Polyhedral models for K-theory 

Leonid Monin

One can associate a commutative, graded algebra which satisfies Poincare duality to a homogeneous polynomial f on a vector space V. One particularly interesting example of this construction is when f is the volume polynomial on a suitable space of (virtual) polytopes. In this case the algebra A_f recovers cohomology rings of toric or flag varieties. 

In my talk I will explain these results and present their recent generalizations. In particular, I will explain how to associate an algebra with Gorenstein duality to any function g on a lattice L. In the case when g is the Ehrhart function on a lattice of integer (virtual) polytopes, this construction recovers K-theory of toric and full flag varieties.

The Kobayashi conjecture for compact hyperkähler varieties 

Christian Lehn

In a joint work with Ljudmila Kamenova, we improve on a work by Kamenova-Lu-Verbitsky by showing the vanishing of the Kobayashi pseudometric for compact hyperkähler manifolds assuming that the second Betti number is at least 7 and the SYZ-conjecture holds. This in particular applies to all known examples of compact hyperkähler manifolds.

Conference dinner in Versilia

Higher Fano Manifolds

Ana-Maria Castravet

Fano manifolds are complex projective manifolds having positive first Chern class. The positivity condition on the first Chern class has far reaching geometric and arithmetic implications. For instance, Fano manifolds are covered by rational curves, and families of Fano manifolds over one dimensional bases always admit holomorphic sections. In recent years, there has been some effort towards defining suitable higher analogues of the Fano condition. Higher Fano manifolds are expected to enjoy stronger versions of several of the nice properties of Fano manifolds. In this talk, I will discuss higher Fano manifolds which are defined in terms of positivity of higher Chern characters. After a brief survey of what is currently known, I will present recent joint work with Carolina Araujo, Roya Beheshti, Kelly Jabbusch, Svetlana Makarova, Enrica Mazzon and Nivedita Viswanathan, regarding toric higher Fano manifolds. I will explain a strategy towards proving that projective spaces are the only higher Fano manifolds among smooth projective toric varieties. 

Self-dual matroids from canonical curves (slides)

Alheydis Geiger

A hyperplane section of a canonically embedded, non-hypereliptic smooth curve of genus g consists of 2g-2 points in g-2 dimensional projective space. From work by Dolgachev and Ortland it is known that point configurations obtained in such a way are self-associated. We interpret this notion in terms of matroids: A generic hyperplane section of a canonical curve gives rise to an identically self-dual matroid. This can also be seen as a combinatorial shadow of the Riemann-Roch theorem. This procedure also works for graph curves, which are stable canonical curves consisting of 2g-2 lines.

Self-dual point configurations are parametrized by a subvariety of the Grassmannian Gr(n,2n) and its tropicalization, which we analyze in detail for n=3. Further, we classify identically self-dual matroids of rank up to five and determine the dimension of their (self-dual) realization spaces. Building on works by Bath, Mukai and Petrakiev, we investigate the question, which self-dual point configurations can be obtained by a hyperplane section of a canonical curve. This project is joined work with Sachi Hashimoto, Bernd Sturmfels and Raluca Vlad.