Schedule for: 17w5130 - Beyond Toric Geometry

A welcome drink will be served at the hotel.

Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities. We extend this picture to log terminal singularities in any dimension coming with a torus action of complexity one. In this setting, the previously finite groups become solvable torus extensions. As explicit examples, we investigate compound du Val threefold singularities. We give a complete classification and exhibit all the possible chains of iterated Cox rings.

we provide a characterization of smooth algebraic varieties endowed with a faithful algebraic torus action in terms of a combinatorial description given by Altmann and Hausen. Our main result is that such a variety X is smooth if and only if it is locally isomorphic in the \'etale topology to the affine space endowed with a linear torus action. Furthermore, this is the case if and only if the combinatorial data describing X is locally isomorphic in the \'etale topology to the combinatorial data describing affine space endowed with a linear torus action.

In this talk we consider the problem of finding an ideal whose blowup defines the Nash blowup of a toric surface and such that its zero locus coincides with the singular set of the toric surface. Our main result gives a positive consequence of removing the hypothesis of normality in the definition of toric varieties.

A quiver is a finite directed graph and a representation of a quiver is an assignment of vector space to each vertex and linear map to each arrow. Once the vector spaces at each vertex have been fixed, the space of representations is an algebraic variety. This variety carries an action of a product of general linear groups, which acts by change of basis. I'll focus on the setting where the quiver's underlying graph is a type A Dynkin diagram, and discuss results on the geometry and combinatorics of the associated orbit closures (a.k.a. quiver loci). In particular, I'll show that each quiver locus is isomorphic, up to smooth factor, to an open subvariety of a Schubert variety, discuss some geometric consequences of this identification, and describe combinatorial formulas for multidegrees and K-polynomials of type A quiver loci. This is joint work with Ryan Kinser and Allen Knutson.

I will describe the geometry of the $2m$-dimensional Fano manifold $G$ parametrizing $(m − 1)$-planes in a smooth complete intersection $Z$ of two quadric hypersurfaces in the complex projective space $\mathbb{P}^{2m+2}$, for $m > 0$. We will see that there are exactly $22m+2$ distinct isomorphisms in codimension one between $G$ and the blow-up $X$ of $\mathbb{P}^{2m}$ at $2m + 3$ general points, parametrized by the $22m+2$ distinct $m$-planes contained in $Z$. The varieties $G$ and $X$ are Mori dream spaces, and the birational maps $G\to X$ allow to determine the Mori chamber decomposition of the cone of effective divisors of G, and the automorphism group of G. This generalizes to arbitrary even dimension the classical description of quartic del Pezzo surfaces $(m = 1)$. This is a joint work with Carolina Araujo (IMPA).

A normal affine toric variety X with no torus factors has a canonical equivariant embedding in affine space, determined by the Hilbert basis of the weight cone of X. This embedding has many nice properties: the tropicalization Trop(X) with respect to this embedding is a linear subspace of R^n, every initial ideal corresponding to a point in Trop(X) is simply the ideal of X, and the generators for the coordinate ring of X form a Khovanskii basis with respect to any full-rank homogeneous valuation. In this talk, I will report on joint work with Nathan Ilten in which we generalize this situation to normal rational affine varieties equipped with the action of a codimension-one torus. We produce explicit affine embeddings of such varieties, and show that these embeddings enjoy properties similar to those found in the toric situation.

To each torus-equivariant vector bundle over a smooth complete toric variety, we associated a representable matroid (essentially a finite collection of vectors).  In this talk, we will describe how the combinatorics of this matroid encodes a resolution of the toric vector bundles by a complex whose terms are direct sums of toric line bundles. With some luck, we will also outline some applications to the equations and syzygies of smooth projective toric varieties.

I will discuss new localization and Riemann-Roch results in equivariant operational K-theory, showing that equiavariant operational K-theory is related to equivariant Chow cohomology through a Riemann-Roch transformation that factors the Chern character. In this framework, the Atiyah-Bott-Berline-Vergne localization formula becomes the analogue of Hirzebruch-Riemann-Roch. Applications include examples of toric varieties whose equivariant K-theory of vector bundles (or perfect complexes) does not surject onto its ordinary K-theory, and the computation of equivariant operational K-theory of spherical varieties in terms of fixed-point data. Joint work with D. Anderson and R. Gonzales.

The tropicalization of a projective toric variety is a topological space that "looks like" the associated polytope. Tropicalizations of subvarieties of a toric variety are polyhedral complexes inside this space.  These encode degenerations of the variety, including toric degenerations.  In this talk I will explain a construction of subchemes of a tropical toric variety, developed with Felipe Rincon based on work of the Giansiracusas.

Maps between toric varieties can be lifted to maps between their Cox rings, if one is allowed to modify the Cox ring of the domain a bit. This was shown by Gavin Brown and Jarosław Buczyński. In a joint work with Elena Martinengo, we note that this idea becomes more natural when considering quotient stacks. We also extend this lifting to maps between Mori dream spaces. In my talk, I present these ideas and further directions.

The celebrated proof of the Hartshorne conjecture by Shigefumi Mori allowed for the study of the geometry of higher dimensional varieties through the analysis of deformations of rational curves. One of the many applications of Mori's results was Lazarsfeld's positive answer to the conjecture of Remmert and Van de Ven which states that the only smooth variety that the projective space can map surjectively onto is the projective space itself. Motivated by this result, a similar problem has been considered for other kinds of varieties such as abelian varieties (Demailly-Hwang-Mok-Peternell) or toric varieties (Occhetta-Wiśniewski). In my talk, I would like to present a completely new perspective on the problem coming from the study of Frobenius lifts in positive characteristic. This is based on a joint project with Jakub Witaszek and Maciej Zdanowicz.

Mori Dream Spaces were introduced by Hu and Keel as normal, Q-factorial projective varieties whose effective cone admits a nice decomposition. Mori's minimal model program can be run for every divisor on a Mori Dream Space. Recently there have been many studies on the question that for which integers a,b,c the blow-up of the weighted projective plane P(a,b,c) at a general point is a Mori Dream Space. In this talk, I will recall these recent work, and introduce a generalization of a result by González and Karu in 2014. Specifically, for some toric surfaces of Picard number one, whether the blow-up is a Mori Dream Space is equivalent to countably many planar interpolation problems. I will give new examples and non-examples of Mori Dream Spaces, along with a conjecture of more non-examples, by reducing these interpolation problems.

We classify holomorphic as well as algebraic $T$-equivariant principal $G$-bundles $E$ over a nonsingular toric variety $X$, where $G$ is a complex linear algebraic group. We will also see that any algebraic principal $G$-bundle $E_G$ on $X$ is $T$-equivariant iff $E_G$ admits a logarithmic connection singular over $X-T$.  This is a joint work with Indranil Biswas and Mainak Poddar.

Let $T$ be a split maximal torus of a semisimple linear algebraic group $G$. Consider the category of $T$-equivariant motives of projective homogeneous $G$-varieties. We describe an endomorphism ring of the $T$-motive of the total $G$-flag using the relative push-pull operators. The talk is based on the results of arXiv:1609.06929.

Chow rings of smooth projective toric varieties admit a convenient functorial description in terms of polytope rings introduced by Khovanskii and Pukhlikov. An analogous description for Chow rings of complete flag varieties was obtained by Kave and used by Kiritchenko, Smirnov and Timorin to get positive presentations of Schubert cycles by faces of a Gelfand-Zetlin polytope in type A. The underlying combinatorics was based on the mitosis of Knutson and Miller in type A. In my talk, I will describe a new mitosis algorithm on faces of a symplectic Gelfand-Zetlin polytope. Conjecturally, the collections of faces produced by this algoritm yield positive presentations of Schubert cycles in type C (joint work with Maria Padalko).

Using reflection groups of the Picard lattice, and (sometime) unique reconstruction of Bott-Samelson varieties, one can characterize total flag varieties of semisimple algebraic groups in terms of their elementary contractions. This implies their rigidity in families of Fano manifolds. The talk is based on a joint projects with Munoz, Occhetta, Sola Conde, Watanabe and Weber.