# Toric Degenerations

## Date and Time

11. June 2021 from 14:00 - 17:30.

## Schedule

• 14:00 - 15:00: Victor Batyrev: Variations on the theme of classical discriminant

• 15:15 - 16:15: Lara Bossinger: Newton--Okounkov bodies for cluster varieties

• 16:30 - 17:30: Chris Manon: When is a (projectivized) toric vector bundle a Mori dream space?

## Abstracts

• Victor Batyrev: Variations on the theme of classical discriminant

Abstract: The classical discriminant $\Delta_n(f)$ of a degree $n$ polynomial $f(x)$ is an irreducible homogeneous polynomial of degree $2n-2$ on the coefficients $a_0, \ldots, a_n$ of $f$ that vanishes if and only if $f$ has a multiple zero. I will explain a tropical proof of the theorem of Gelfand, Kapranov and Zelevinsky (1990) that identifies the Newton polytope $P_n$ of $\Delta_n$ with a $(n-1)$-dimensional combinatorial cube obtained from the classical root system of type $A_{n-1}$. Recently Mikhalkin and Tsikh (2017) discovered a nice factorization property for truncations of $\Delta_n$ with respect to facets $\Gamma_i$ of $P_n$ containing the vertex $v_0 \in P_n$ corresponding to the monomial $a_1^2 \cdots a_{n-1}^2 \in \Delta_n$. I will give a GKZ-proof of this property and show its connection to the boundary stata in the $(n-1)$-dimensional toric Losev-Manin moduli space $\overline{L_n}$. Some variations on the above statements will be discussed in connection to the toric moduli space associated with the root system of type $B_n$ and the mirror symmetry for $3$-dimensional cyclic quotient singularities ${\mathbb C}^3/\mu_{2n+1}$.

• Lara Bossinger: Newton--Okounkov bodies for cluster varieties

Abstract: Cluster varieties are schemes glued from algebraic tori. Just as tori themselves, they come in dual pairs and it is good to think of them as generalizing tori. Just as compactifications of tori give rise to interesting varieties, (partial) compactifications of cluster varieties include examples such as Grassmannians, partial flag varieties or configurations spaces. A few years ago Gross--Hacking--Keel--Kontsevich developed a mirror symmetry inspired program for cluster varieties. I will explain how their tools can be used to obtain valuations and Newton--Okounkov bodies for their (partial) compactifications. The rich structure of cluster varieties however can be exploited even further in this context which leads us to an intrinsic definition of a Newton--Okounkov body.

The theory of cluster varieties interacts beautifully with representation theory and algebraic groups. I will exhibit this connection by comparing GHKK's technology with known mirror symmetry constructions such as those by Givental, Baytev--Ciocan-Fontanini--Kim--van Straten, Rietsch and Marsh--Rietsch.

• Chris Manon: When is a (projectivized) toric vector bundle a Mori dream space?

Abstract: Like toric varieties, toric vector bundles are a rich class of varieties which admit a combinatorial description. Following the classification due to Klyachko, a toric vector bundle is captured by a subspace arrangement decorated by toric data. This makes toric vector bundles an accessible test-bed for concepts from algebraic geometry. Along these lines, Hering, Payne, and Mustata asked if the projectivization of a toric vector bundle is always a Mori dream space. Suess and Hausen, and Gonzales showed that the answer is "yes" for tangent bundles of smooth, projective toric varieties, and rank 2 vector bundles, respectively. Then Hering, Payne, Gonzales, and Suess showed the answer in general must be "no" by constructing an elegant relationship between toric vector bundles and various blow-ups of projective spaces, in particular the blow-ups of general arrangements of points studied by Castravet, Tevelev and Mukai. In this talk I will review some of these results, and then give a new description of toric vector bundles by tropical information. This description allows us to characterize the Mori dream space property in terms of tropical and algebraic data, and produce new families of Mori dream spaces indexed by the integral points in a locally closed polyhedral complex. Along the way I'll discuss plenty of examples and some questions. This is joint work with Kiumars Kaveh.

## Location

Via Zoom. Please register with an informal email to one of the organizers in order to receive the link.