A Focused Research Workshop on Toric Degenerations

21 August 2017, 1.00 PM - 24 August 2017, 12.00 PM

Bristol, UK

Organiser


Speakers and Schedule

 
The talks will be at the 4th Floor Seminar Room in Howard House. Each talk is scheduled for 45-50 minutes followed by 10 minutes for questions.
 

Monday, August 21

 
13:00-14:00 Diane Maclagan (Warwick) Tree compactifications of the moduli space of genus zero curves
14:00-15:00 Alex Fink (Queen Mary) Stiefel tropical linear spaces
15:00-15:30 Coffee Break
15:30-16:30 Elisa Postinghel (Loughborough) Toric degenerations of Mori dream spaces via tropical compactifications
16:30-17:30 Bernt Ivar Utstøl Nødland (Oslo) Local Euler obstructions of toric varieties
18:00 Social Dinner

Tuesday, August 22

 
9:30-10:30 Robert Marsh (Leeds) Twists of Pluecker coordinates as dimer partition functions
10:30-11:00 Coffee Break
11:00-12:00 Lars Halvard Halle (Copenhagen) Motivic zeta functions of degenerating Calabi-Yau varieties
12:00-14:00 Lunch break
14:00-15:00 Volkmar Welker (Marburg) The Grassmann Associahedron
15:00-15:30 Coffee Break
15:30-16:30 Xin Fang (Köln) Semi-toric degenerations of Grassmanians arising from Newton-Okounkov bodies
16:30-17:30 Nelly Villamizar (Swansea) Varieties of apolar subschemes of toric surfaces
18:00 Social Dinner

Wednesday, August 23

9:30-10:30 Milena Hering (Edinburgh) Frobenius splittings of toric varieties and unimodularity
10:30-11:00 Coffee Break
11:00-12:00 Martina Lanini (Rome) Cohomology of the flag variety under PBW degenerations
12:00-14:00 Lunch break
14:00-15:00 Emilie Dufresne (Nottingham) Mapping toric varieties into small dimensional spaces
15:00-15:30 Coffee Break
15:30-16:30 Lara Bossinger (Köln) Birational sequences for Grassmannian and trop(Gr(2,n))
16:30-17:30 Takuya Murata (Pittsburgh) A toric degeneration through symbolic normal cones
18:00 Social Dinner

Thursday, August 24

 
9:00-10:00 Kristin Shaw (MPI, Leipzig) Toric degenerations and Khovanskii bases of Grassmannians
10:00-10:15 Coffee Break
10:15-11:15 Christopher Manon (George Mason) Constructing Khovanskii bases in representation theory
11:15-12:15 Kalina Mincheva (Yale University) Computing toric degenerations of flag varieties

Participants


Abstracts of Talks

    • Lara Bossinger (Köln)
    • Title: Birational sequences for Grassmannian and trop(Gr(2,n))
      Abstract: Birational sequences have been recently introduced by Fang, Fourier, and Littelmann. The provide a unified construction for toric degenerations of flag varieties (and more general spherical varieties) from representation theory. For example, Caldero's construction and the one of Alexeev-Brion are special cases. I will present a class of binational sequences, called iterated, that provide toric degenerations of Grassmannians. I will further show that for every degeneration of the Grassmannian of planes constructed using the tropicalization there exists an iterated birational sequences yielding an isomorphic degeneration.

    • Emilie Dufresne (Nottingham)
    • Title: Mapping toric varieties into small dimensional spaces
      Abstract: A smooth d-dimensional projective variety X can always be embedded into 2d + 1-dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax our requirement and ask only that the map is injective, then any d-dimensional projective (resp. affine) variety can be mapped injectively to 2d + 1-dimensional projective space (resp. affine). A natural question then arises: what is the minimal m such that a projective variety can be mapped injectively to m-dimensional projective space? In this talk I discuss this question for the afffine cones over normal toric varieties, with the most complete results being for the affine cones over Segre-Veronese varieties.
      Joint work with Jack Jeffries.

    • Xin Fang (Köln)
    • Title: Semi-toric degenerations of Grassmanians arising from Newton-Okounkov bodies
      Abstract: The standard monomial theory of Pluecker algebras, which dates back to Hodge, provides monomial bases of these algebras. Geometrically, these bases can be applied to degenerate the Grassmann variety to a reduced union of toric varieties, which are called semi-toric varieties. In this talk I will explain how these constructions can be combined with Newton-Okounkov bodies and the associated toric degenerations.
      This talk is based on joint work with Peter Littelmann.

    • Alex Fink (Queen Mary)
    • Title: Stiefel tropical linear spaces
      Abstract: Unlike the classical situation, not all tropical linear spaces are row spaces of matrices, i.e. stable sums of points. To the ones which are we give the name _Stiefel tropical linear spaces_. We discuss their combinatorial structure viewed through the lenses of transversal matroids or of tropical parametrised images, and a nice atlas of local coordinate systems for their moduli space. The construction has also been used to define a class of "principal" tropical ideals.
      This talk is based on joint work with Felipe Rincon.

    • Lars Halvard Halle (Copenhagen)
    • Title: Motivic zeta functions of degenerating Calabi-Yau varieties
      Abstract: Let K = C((t)), and let X be a smooth projective K-variety with trivial canonical sheaf. To X, one can associate its so-called motivic zeta function Z_X(T). This is a formal power series in T which encodes the asymptotic behaviour of the set of rational points of X under ramified extension of K. The properties of this series are closely related to the behaviour of X under degeneration. I will discuss recent joint work with J. Nicaise, where we investigate the case where $X$ admits a particularly nice type of model (called equivariant Kulikov model), after some suitable base change.

    • Milena Hering (Edinburgh)
    • Title: Frobenius splittings of toric varieties and unimodularity
      Abstract: Many nice properties of toric varieties can be deduced from the fact that they admit a splitting of the Frobenius morphism. Varieties whose product with themselves admits a splitting compatible with the diagonal exhibit even nicer properties. For example, every ample line bundle on such a variety is normally generated. In this talk I will give an overview over the consequences of the existence of such splittings and discuss new combinatorial criteria for toric varieties to be diagonally split.

    • Martina Lanini (University of Rome)
    • Title: Cohomology of the flag variety under PBW degenerations
      Abstract: Given a complex algebraic variety and a flat deformation of if, there is an induced homomorphism from the cohomology of the latter to the cohomology of the original variety. Such a homomorphism can in general fail to be surjective or injective. In my talk, I will discuss recent work with Elisabetta Strickland, in which we show that in the case of PBW degenerations of a (type A) flag variety, the induced homomorphism between cohomologies is surjective. PBW degenerations have been recently introduced by Cerulli Irelli, Fang, Feigin, Fourier and Reineke, and proven to have the nice property to be isomorphic to Schubert varieties. Our result shows once more that these degenerations are extremely well behaved.

    • Robert Marsh (Leeds)
    • Title: Twists of Pluecker coordinates as dimer partition functions
      Abstract: The homogeneous coordinate ring of the Grassmannian has a cluster algebra structure defined in terms of planar diagrams known as Postnikov diagrams. In particular, it is generated by the union of overlapping subsets (clusters) of equal cardinality, related by mutations. The cluster corresponding to a Postnikov diagram consists entirely of Pluecker coordinates. We introduce a twist map on the Grassmannian related to the Berenstein-Zelevinsky twist, and give an explicit Laurent expansion for the twist of an arbitrary Pluecker coordinate, in terms of the cluster variables associated with a fixed Postnikov diagram. The expansion arises as a (scaled) dimer partition function (i.e. matching polynomial) of a weighted version of the bipartite graph dual to the Postnikov diagram, modified by a boundary condition determined by the Pluecker coordinate.
      Joint work with J. S. Scott.

    • Diane Maclagan (Warwick)
    • Title: Tree compactifications of the moduli space of genus zero curves
      Abstract: The moduli space M_{0,n} of smooth genus zero curves with n marked points has a standard compactification by the Deligne-Mumford module space of stable genus zero curves with n marked points. The compactification can be constructed as the closure of M_{0,n} inside a toric variety. The fan of the toric variety is moduli space of phylogenetic trees. I will discuss joint work with Dustin Cartwright to construct other compactifications of M_{0,n} by varying the toric variety by using variants of phylogenetic trees. These compactifications include many of the standard alternative compactifications of M_{0,n}.

    • Kalina Mincheva (Yale University)
    • Title: Computing toric degenerations of flag varieties.
      Abstract: We compute toric degenerations arising from the tropicalization of the complete flag of GL(4) and GL(5). In these cases we compare toric degenerations arising from string polytopes and the FFLV-polytope with those obtained from the tropicalization of the flag varieties. We present a general procedure to find such degenerations even in the cases where the initial ideal arising from a cone of the tropicalization is not prime.
      Joint work with Lara Bossinger, Sara Lamboglia and Fatemeh Mohammadi.

    • Takuya Murata (University of Pittsburgh)
    • Title: a toric degeneration through symbolic normal cones
      Abstract: I explain some of techniques used in my pre-print with Kiumars Kaveh: https://arxiv.org/abs/1708.02698 The pre-print uses a variant of degeneration to a normal cone, called a degeneration to a symbolic normal cone (the notion inspired by Knutson's balanced normal cone.) The key property of this variation is that a symbolic normal cone along a certain set-theoretic-hypersurface is a variety; i.e., reduced, irreducible, finite-type. Now, starting from a polarized projective variety X = Proj(R), we degenerate the affine cone Spec(R) step-by-step; so in the end, we get a sequence of flat degenerations to a projective non-normal toric variety that preserves polarizations up to Veronese embeddings. The geometric invariant theory and Bertin's theorem guarantee there is a sufficient supply of hypersurfaces for this induction scheme to be used.

    • Elisa Postinghel (Loughborough)
    • Title: Toric degenerations of Mori dream spaces via tropical compactifications
      Abstract: Let X be a Mori dream space embedded in a toric variety with algebraic torus T. We will construct a tropical compactification of the restriction of X to T that determines a model of X dominating all its small Q-factorial modifications. Exploiting the combinatorial properties of such compactification, we can show that there exist maximal rank valuations whose corresponding value semigroups are finitely generated, and hence each of them determines a (not necessarily normal) toric variety. Using work of Anderson, we can construct degenerations of X to these toric varieties. During the talk we shall look at some familiar examples, such as del Pezzo surfaces and certain moduli spaces.
      This is based on joint work with Stefano Urbinati.

    • Kristin Shaw (Max Planck Institute, Leipzig)
    • Title: Toric degenerations and Khovanskii bases of Grassmannians
      Abstract: Many toric degenerations and integrable systems of the Grassmannians Gr(2, n) are described by trees, or equivalently subdivisions of polygons. These degenerations can also be seen to arise from top dimensional cones of the tropicalisation of the Grassmannian. In this talk we study the situation of initial degenerations of higher Grassmannians with respect to weight vectors in their tropicalisations. We focus on particular combinatorial types of cones in tropical Grassmannians which correspond to matching fields. From here we define matching field ideals and prove a necessary condition for such an initial degeneration to be toric. Moreover, in this setting the Pluecker relations form a Khovanskii basis. We also show that this condition is sufficient in the case of Gr(3,n) provided that the matching field ideal is generated in degree two. This leads to many combinatorial conjectures in relation to matching field ideals.
      This is joint work with Fatemeh Mohammadi.

    • Bernt Ivar Utstøl Nødland (Oslo)
    • Title: Local Euler obstructions of toric varieties
      Abstract: Any projective variety has an associated dual variety, which typically will be a hypersurface. For smooth toric varieties a formula for the degree of this hypersurface is given by Gelfand, Kapranov and Zelevinsky. Matsui and Takeuchi generalized this to arbitrary toric varieties, where the key extra invariant needed is the local Euler obstruction. They give a formula to compute its value on a given T-orbit, which is recursive in the codimension of the orbit. In this talk I will discuss how to calculate these invariants for toric surfaces and threefolds.

    • Nelly Villamizar (Swansea)
    • Title: Varieties of apolar subschemes of toric surfaces
      Abstract: The variety of sums of powers associated to a homogeneous polynomial describes the additive decompositions of the polynomial into powers of linear forms. These polynomial decompositions appear in several areas of application, they are strongly connected to questions in representation theory, coding theory, signal processing, data analysis, and algebraic statistics. One of the most useful tools to study varieties of sums of powers is apolarity, a notion originally related to the action of differential operators on the polynomial ring. This notion can be generalized in terms of the Cox ring of a variety, and in this way varieties of sum of powers are a special case of varieties of apolar schemes. In the talk, I will present this generalization and examples of such varieties in the case of toric surfaces, when the Cox ring is particularly well-behaved.
      The results are part of our work with Kristian Ranestad and Matteo Gallet on this topic.

    • Volkmar Welker (Marburg)
    • Title: The Grassmann Associahedron
      Abstract: We consider the coordinate ring of the Grassmannian G(n,r) of r-planes in C^n as a quotient of the polynomial ring over the Pl"ucker coordinates by the ideal I(n,r). Using a toric degeneration of I(n,r) it follows from work of Reiner and Welker, that I(n,r) has an initial ideal that is the Stanley-Reisner ring of a simplical polytope. Their polytopes do not seem to be "natural". Then work of Jonsson and Welker for r=2, Petersen, P. Pylyavskyy, and D. E. Speyer constructed a "natural" choice, called Grassmann Associahedron. We report on work with Santos and Stump where we give an alternative construction and exhibit combinatorial features of the polytope.