Convexity Workshop: Titles and Abstracts

Indecomposable commutative algebraic groups

by Michel Brion (Institut Fourier)
Friday, 7 October 2016 at 11:00 in Room 230 at the Fields Institute

Commutative algebraic groups occur in several areas of algebraic geometry, e.g. as Jacobians of (possibly singular) curves. Their structure is not so well understood in general, and becomes much simpler when considering commutative algebraic groups up to isogeny. For example, every such group has a unique decomposition into indecomposable summands in this setting. The talk will address the category of commutative algebraic groups up to isogeny and the classification of its indecomposable objects, which can be carried out in certain cases by means of valued graphs.

Some new four-dimensional Fano manifolds

by Tom Coates (Imperial)
Tuesday, 4 October 2016 at 9:00 in Room 230 at the Fields Institute

Fano manifolds are basic building blocks in algebraic geometry. There is exactly 1 one-dimensional Fano manifold: the line. There are 10 deformation families of two-dimensional Fano manifolds: the del Pezzo surfaces, known since the 19th century. There are 105 deformation families of three-dimensional Fano manifolds: these were classified by Mori–Mukai in the 1990s, as a spectacular application of Mori theory. Very little is known about Fano classification in dimension four or more.

I will describe work by Kalashnikov and joint work with Kasprzyk and Prince that constructs several hundred new four-dimensional Fano manifolds, as subvarieties of quiver flag manifolds and toric varieties. I will explain how this fits in to a program — joint work with Corti, Galkin, Golsyhev, Kasprzyk, and others — to find and classify Fano manifolds using mirror symmetry.

Anticanonical tropical del Pezzo cubic surfaces contain exactly 27 lines

by Maria Angelica Cueto (Ohio State)
Friday, 7 October 2016 at 14:00 in Room 230 at the Fields Institute

Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement “any smooth surface of degree 3 in P^3 contains exactly 27 lines” is known to be false tropically. Work of Vigeland from 2007 provides examples of cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.

In this talk I will explain how to correct this pathology. The novel idea is to consider the embedding of a smooth cubic surface in P^44 via its anticanonical bundle. The tropicalization induced by this embedding contains exactly 27 lines under a mild genericity assumption. More precisely, smooth cubic surfaces in P^3 are del Pezzos, and can be obtained by blowing up P^2 at six points in general position. We identify these points with six parameters over a field with nontrivial valuation. Our genericity assumption involves the valuations of 36 linear expressions in these parameters which give the positive roots of type E_6. Tropical convexity plays a central role in ruling out the existence of extra tropical lines on the tropical cubic surface.

This talk is based on an ongoing project joint with Anand Deopurkar.

Convex visualisation of toric vector bundles

by Sandra Di Rocco (KTH)
Wednesday, 5 October 2016 at 9:00 in Room 230 at the Fields Institute

Many properties of line bundles and vector bundles of higher rank become evident from the convex polytopes associated to their global sections. If the vector bundle has rank higher than one certain useful and nice equivalences are no longer valid. This talk will recall the dictionary between convex geometry and positivity in algebraic geometry, available for line bundles, and present the corresponding picture for higher rank toric vector bundles. The higher rank results are joint work with K. Jabbusch and G.G. Smith.

Convexity and categorification in Boij-Soederberg theory

by Daniel Erman (Wisconsin)
Wednesday, 5 October 2016 at 10:15 in Room 230 at the Fields Institute

I’ll discuss the convex cones that arise in Boij-Soederberg theory, and how those results lead to deeper—as yet unresolved—structural questions about vector bundles.

Enumeration of points, lines, planes, etc.

by June Huh (Princeton)
Friday, 7 October 2016 at 16:00 in Room 230 at the Fields Institute

One of the earliest results in enumerative combinatorial geometry is the following theorem of de Brujin and Erdős: Every set of points E in a projective plane determines at least |E| lines, unless all the points are contained in a line. Motzkin and others extended the result to higher dimensions, who showed that every set of points E in a projective space determines at least |E| hyperplanes, unless all the points are contained in a hyperplane. Let E be a spanning subset of a d-dimensional vector space. We show that, in the poset of subspaces spanned by subsets of E, there are at least as many (d-k)-dimensional subspaces as there are k-dimensional subspaces, for every k at most d/2. This confirms the “top-heavy” conjecture of Dowling and Wilson for all matroids realizable over some field. The proof relies on the decomposition theorem package for l-adic intersection complexes. Joint work with Botong Wang.

K-Stability for Fano varieties with torus action

by Nathan Ilten (Simon Fraser)
Tuesday, 4 October 2016 at 10:45 in Room 230 at the Fields Institute

It has been recently shown by Chen-Donaldson-Sun that the existence of a Kähler-Einstein metric on a Fano manifold is equivalent to the property of K-stability. In general, however, this does not lead to an effective criterion for deciding whether such a metric exists, since verifying the property of K-stability requires one to consider infinitely many special degenerations called test configurations. I will discuss  joint work with H. Süß in which we show that for Fano manifolds with complexity-one torus actions, there are only finitely many test configurations one needs to consider. This leads to an effective method for verifying K-stability, and hence the existence of a Kähler-Einstein metric. As an application, we provide new examples of Kähler-Einstein Fano threefolds. 

Okounkov bodies and tropical geometry

by Eric Katz (Ohio State)
Monday, 3 October 2016 at 9:30 in Room 230 at the Fields Institute

Tropical geometry and Okounkov bodies are generalizations of the theory of Newton polytopes in different directs: tropical geometry for higher codimensions; Okounkov bodies for non-toric ambient spaces. In this talk, we will discuss joint work with Stefano Urbinati which finds these two theories converging again. We first discuss the analogue of Okounkov bodies over discrete valuation rings, using constructions motivated both by tropical geometry and Arakelov theory. In the special case of semistable families of curves, the theory of linear systems on graphs makes an appearance. This gives some pointers to a higher dimensional theory of combinatorial linear systems.

Grobner theory and tropical geometry on spherical varieties

by Kiumars Kaveh (Pittsburgh)
Tuesday, 4 October 2016 at 15:00 in Room 230 at the Fields Institute

Let G be a connected reductive algebraic group. I will talk about a Grobner theory for multiplicity-free G-algebras, as well as a tropical geometry for subschemes in a spherical G-homogeneous space G/H. We will discuss the notions of a spherical tropical variety and a fundamental theorem of tropical geometry in this context. We also propose a definition for a spherical amoeba in G/H and talk about the principle that amoeba approaches the tropical variety. This is directly related to the (Archimedean) Cartan decomposition for G/H. A particular case of this states that “invariant factors” of a matrix (over Laurent series) are a limit of its “singular values”. This is a joint work with Chris Manon and builds on the recent work of Tassos Vogiannou.

Newton-Okounkov polytopes of Bott-Samelson varieties as Minkowski sums

by Valentina Kiritchenko (HSE)
Monday, 3 October 2016 at 14:00 in Room 230 at the Fields Institute

An essential feature of Newton polytopes of polarized toric varieties is the additivity property with respect to the Minkowski sum, that is, tensor product of line bundles corresponds to the Minkowski sum of Newton polytopes. In particular, this property is crucial for the famous Bernstein-Koushnirenko theorem. The additivity property does not necessarily hold for Newton-Okounkov convex bodies of more general varieties and valuations. We show that the additivity property holds for a geometric valuation on a Bott-Samelson resolution
of the variety of complete flags in C^n. The resulting Newton-Okounkov polytopes are combinatorially different from previously known polytopes, and can be obtained as Minkowski sums of Newton-Okounkov polytopes of varieties of complete flags in C^2,…, C^n.

Syzygies of abelian surfaces, construction of singular divisors, and Newton-Okounkov bodies

by Alex Küronya (Frankfurt)
Monday, 3 October 2016 at 16:00 in Room 230 at the Fields Institute

Constructing divisors with prescribed singularities is one of the most powerful techniques in modern projective geometry, leading to proofs of major results in the minimal model program and the strongest general positivity theorems by Angehrn-Siu and Kollár-Matsusaka. We present a novel method for constructing singular divisors on surfaces based on infinitesimal Newton-Okounkov bodies. As an application of our machinery we discuss a Reider-type theorem for higher syzygies on abelian varieties building on earlier work of Lazarsfeld-Pareschi-Popa.

Canonical forms for polytopes and amplituhedra

by Thomas Lam (Michigan)
Thursday, 6 October 2016 at 9:00 in Room 230 at the Fields Institute

A projective polytope comes equipped with a remarkable rational top form with simple poles along the boundary of the polytope. I will explain some properties and constructions of this canonical form. I will then discuss a conjectural extension of this differential form to polytopes inside Grassmannians, which is motivated by recent work in physics. This talk is based on joint work with Nima Arkani-Hamed and Yuntao Bai.

Degenerate flags and Schubert varieties

by Martina Lanini (Edinburgh)
Friday, 7 October 2016 at 9:30 in Room 230 at the Fields Institute

Introduced in 2010 by Evgeny Feigin, degenerate flag varieties are degenerations of flag manifolds, naturally arising from a representation theoretic context. In this talk, I will discuss joint work with G. Cerulli Irelli, and G. Cerulli Irelli and P. Littelmann, in which we show that such degenerations in type A and C not only share a lot of properties with Schubert varieties (as previously proven by Feigin, Finkelberg and Littelmann), but are in fact Schubert varieties in an appropriate flag manifold.

Newton-Okounkov bodies, Khovanskii bases, and tropical geometry
of projective varieties

by Chris Manon (George Mason)
Monday, 3 October 2016 at 11:00 in Room 230 at the Fields Institute

The theory of Newton-Okounkov bodies assigns a convex set to a choice of a maximal rank valuation on a graded algebra. When this algebra is taken to be a section ring of a projective variety $X$, and the valuation satisfies a few extra conditions, this construction can be used to define a flat degeneration from $X$ to a projective toric variety. I will describe joint work with Kiumars Kaveh where we use tropical geometry to give a necessary and sufficient condition for the existence of such a valuation on a coordinate ring of a projective variety. I also discuss methods for explicitly computing this valuation, and illustrate the construction on a few examples.

Scissors rings for polyhedra and geometric invariants of semialgebraic sets

by Sam Payne (Yale)
Thursday, 6 October 2016 at 15:00 in Room 230 at the Fields Institute

I will discuss tropical methods for computing geometric invariants of semialgebraic sets, defined by polynomial equations and inequalities on valuations over an algebraically closed valued field, through relations to motivic integration in the sense of Hrushovski-Kazhdan, in which the universal invariant is expressed in terms of a tensor product of a scissors ring of polyhedra with a Grothendieck ring of varieties over the residue field.  Based on joint work with Nicaise and Schroeter.

Rational quartic spectrahedra

by Kristian Ranestad (Olso)
Tuesday, 4 Octobert 2016 at 16:15 in Room 230 at the Fields Institute

Rational quartic spectrahedra in real 3-space are semialgebraic convex subsets of semidefinite 4×4 real symmetric matrices, whose boundary admits a rational parameterization. The Zariski closure in complex projective space of the boundary is a symmetroid. If the symmetroid have only simple double points as singularities, it is irrational, in fact birational to a K3-surface, so rational symmetries are special. Rational quartic symmetroids have a a triple point, an elliptic double point or is singular along a curve. Reporting on common work in progress with Martin Helsøe, I shall give several examples and first results towards a classification.

Tropical geometry via logarithmic geometry

by Helge Ruddat (Mainz)
Thursday, 6 October at 16:15 in Room 230 at the Fields Institute

I will survey how tropicalizations in logarithmic geometry lead to a natural generalization of the classical tropical geometry known for toric varieties and their subvarieties. I will mention recent applications of this in log Gromov-Witten theory and results on canonical Calabi-Yau families constructed from convex data.

Irrational Toric Varieties

by Frank Sottile (Texas A&M)
Thursday, 6 October 2016 at 13:30 in Room 230 at the Fields Institute

Classical toric varieties come in two flavours:  Normal toric varieties are given by rational fans in R^n.  A (not necessarily normal) affine toric variety is given by finite subset A of Z^n. When A is homogeneous, it is projective.   Applications of mathematics have long studied the positive real part of a toric variety as the main object, where the points A may be arbitrary points in R^n.  For example, in 1963 Birch showed that such an irrational toric variety is homeomorphic to the convex hull of the set A.

Recent work showing that all Hausdorff limits of translates of irrational toric varieties are toric degenerations suggested the need for a theory of irrational toric varieties associated to arbitrary fans in R^n.  These are R^n_>-equivariant cell complexes dual to the fan.  Among the pleasing parallels with the classical theory is that the space of Hausdorff limits of the irrational projective toric variety of a finite set A in R^n is homeomorphic to the secondary polytope of A.

This talk will sketch this story of irrational toric varieties. It represents work with Garcia-Puente, Zhu, Postinghel, and Villamizar, and work in progress with Pir.

Tropical linear spaces, tropical Grassmannians and matroid subdivisions

by David Speyer (Michigan)
Thursday, 6 October 2016 at 10:45 in Room 230 at the Fields Institute

The Grassmannian parametrizes linear spaces in an n-dimensional vector space. In the same way, the tropicalization of the Grassmannian parameterizes topical linear space in R^n. The combinatorics describing tropical linear spaces leads to studying subdivisions of matroid polytopes into smaller matroid polytopes. This in turn reveals surprising new invariants of matroids.

This talk will attempt to survey work by many authors, including Kapranov; Ardila; Ardila and Klivans; Fink and Derksen; Hacking, Keel and Tevelev as well as the my own work together with Sturmfels and with Fink.

Spherical tropicalization

by Jenia Tevelev (UMass Amherst)
Tuesday, 4 October 2016 at 13:30 in Room 230 at the Fields Institute

I will survey recent results which generalize tropicalization and tropical compactification from subvarieties of an algebraic torus to subvarieties in spherical homogeneous spaces. Most of these results where obtained by Tassos Vogiannou in his PhD thesis.

Free complexes on smooth toric varieties

by Christine Berkesch Zamaere (Minnesota)
Wednesday, 5 October 2016 at 11:30 in Room 230 at the Fields Institute

Given a module M over the Cox ring S of a smooth toric variety, one can consider free complexes that are acyclic modulo irrelevant homology, which we call a complex a free Cox complex for M. These complexes have many advantages over minimal free resolutions over smooth toric varieties other than projective spaces. We develop this in detail for products of projective spaces. This is joint work with Daniel Erman and Gregory G. Smith.