Schedule of talks:

See also the Arithmetic Geometry Seminar

Abstracts

(Li): Berenstein and Zelevinsky introduced quantum cluster algebras and their triangular bases. The support conjecture in [Lee-Li-Rupel-Zelevinsky 2014] gives a conjectural description of the support of a triangular basis element for any rank-2 cluster algebra. In this talk, we explain how to prove the support conjecture for skew-symmetric rank-2 cluster algebras by applying the BBD decomposition theorem to various morphisms related to Nakajima's graded quiver varieties.

(Binder): Tropical homology and cohomology arose as a way to study the Hodge structure of a family of complex projective varieties in passing to a tropical limit. For tropical varieties coming from complex hyperplane arrangement complements, tropical cohomology allows one to compute the de Rham cohomology of the hyperplane arrangement complement in a combinatorial way. As a cohomology theory germane to tropical geometry, tropical cohomology is interesting to study in its own right, with analogues of Poincar´ duality and the Lefschetz (1,1)-Theorem. In this expository talk, I will introduce tropical cohomology, the various ways of computing cohomology, and some interesting properties. Throughout the talk I will give examples and specific computations.

(Yahl): The problem of enumerating linear spaces of a fixed dimension on a variety is known as a Fano problem. Those Fano problems with finitely many solutions have an associated Galois group that acts on the set of solutions. For Fano problems of moderate size with as yet undetermined Galois group, computational methods are used to prove the Galois group is the full symmetric group.

(Dennin): The principal specialization of a Schubert polynomial ν_w := S_w(1,...,1) is known to give the degree of the matrix Schubert variety corresponding to the permutation w. In this talk, we discuss a pattern containment formula (originally conjectured by Yibo Gao) which gives a lower bound for ν_w whenever w is 1243-avoiding. We then see how to extend this result to principal specializations of β-Grothendieck polynomials ν^(β)_w := G^(β)_w(1,...,1) in the setting where w is additionally vexillary. Our methods are bijective, utilizing diagrams called bumpless pipe dreams to obtain combinatorial interpretations of the coefficients c_w and c^(β)_w appearing in these bounds.

(Yong):

(Tsang): Motivated by definitions in mixed Hodge theory, we define the weight filtration and the monodromy weight filtration on the combinatorial intersection cohomology of a fan. These filtrations give a natural definition of the multivariable invariants of subdivisions of polytopes, lattice polytopes and fans, namely the mixed h-polynomial, the refined limit mixed h^*-polynomial, and the mixed cd-index, defined by Katz--Stapledon and Dornian-Katz-Tsang. Previously, only the refined limit mixed h^*-polynomial had a geometric interpretation, which came from filtrations on the cohomology of a schön hypersurface.

(Jensen): Segerman's 15+4 puzzle is a hinged version of the classic 15-puzzle, in which the tiles can rotate as they slide around. In 1974, Wilson classified the groups of solutions to sliding block puzzles. In this talk, we generalize Wilson's result to puzzles like the 15+4 puzzle, where the tiles can rotate, and the sets of solutions are subgroups of the generalized symmetric groups. Aside from two exceptional cases, we will see that the group of solutions to such a puzzle is always either the entire generalized symmetric group or one of two special subgroups of index two.

(George): A projective, normal variety X is called a Mori dream space (MDS) when its Cox ring, Cox(X), is finitely generated. While Mori dream spaces exhibit nice behavior, no complete classification of them is known. Due to their combinatorial nature, one natural class of candidates for Mori dream spaces is projectivized toric vector bundles. In 2012, Jose Gonzalez proved that all rank-2 projectivized toric vector bundles are MDS but complete conditions on rank-r bundles being MDS are still unknown. Kaveh and Manon (2019) gave a combinatorial description of toric vector bundles that we use to describe a family of rank-r toric vector bundles that are MDS. This description, along with a relationship with toric full flag bundles, also allows us to describe conditions under which a direct sum of MDS bundles are also MDS. We conclude with computational examples of bundles over products of projective space and directions for future research, including an algorithmic implementation.

(Sankar): Conic bundles are a geometrically rich class of varieties. In the 70's, Beauville showed that over an algebraically closed field, the group of algebraically trivial curve classes on a conic bundle threefold is isomorphic to the Prym variety of a double cover naturally associated with it. In joint work with Sarah Frei, Lena Ji, Bianca Viray and Isabel Vogt, we study curve classes on certain conic bundle threefolds over arbitrary fields of odd characteristic. We then use the description of these classes to study the rationality of such varieties. Indeed, Hassett-Tschinkel and Benoist-Wittenberg introduced an obstruction to rationality, namely the intermediate Jacobian torsor obstruction, closely related to the structure of the group of curve classes on threefolds. We show that this obstruction is insufficient to characterize rationality.

(Genlik):Gromov-Witten invariants are (virtual) enumerative invariants that play important roles in algebraic geometry and mathematical physics. There is an important class in Gromov-Witten theory; namely, the virtual fundamental class that plays the desired role of the fundamental class. Kontsevich and Manin defined cohomological field theories (CohFTs) to capture the formal properties of the virtual fundamental class in Gromov-Witten theory. If a CohFT is semisimple, then it can be constructed from its topological part. This is known as Givental-Teleman classification of semisimple CohFTs. In this talk, starting from a basic enumerative question, we will describe Gromov-Witten invariants. Then, we will give definition of a CohFT and explain Givental-Teleman classification of semisimple CohFTs. This talk should be accessible to a general audience, and it will serve as a preparation to speaker's next talk.

(Genlik): Physics approach to higher genus mirror symmetry predicts that Gromov-Witten potential of a Calabi-Yau threefold should satisfy certain partial differential equations; namely, the holomorphic anomaly equations. Recently, by works of Lho-Pandharipande, these equations are mathematically proved for some Calabi-Yau threefolds. One such example is C³/Z₃. We generalized this example and proved holomorphic anomaly equations for Cⁿ/Z_n for n greater than or equal to 3, which is a result beyond the consideration of physicists. This is a joint work in progress with Hsian-Hua Tseng.

(Cueto): Splice type surface singularities were introduced by Neumann and Wahl as a generalization of the class of Pham-Brieskorn-Hamm complete intersections of dimension two. Their construction depends on a weighted graph with no loops called a splice diagram. In this talk, I will report on joint work with Patrick Popescu-Pampu and Dmitry Stepanov (arXiv:2108.05912) that sheds new light on these singularities via tropical methods. I will discuss how to reprove some of Neumann and Wahl's earlier results on these singularities, and show that splice type surface singularities are Newton non-degenerate in the sense of Khovanskii.

(Banerjee): We classify Cayley-Bacharach sets in Pⁿ under certain assumptions. Furthermore, we show that Cayley-Bacharach sets lie on low degree curves in Pⁿ, establishing conjectures of Picoco, and Levinson-Ullery under certain simplifying assumptions. As a corollary of our theorems on Cayley-Bacharach sets, we prove results on the convergence of the motive of discriminant complements to a special value of a motivic zeta function. We identify a leading error term for this convergence.

(Wu): Log Fano cone singularities are generalizations of cones over log Fano varieties, and have a local K-stability theory extending the one for log Fano varieties. In this talk, we aim to give a characterization for local K-stability from a non-Archimedean point of view. This characterization will in particular allow us to deal with a more general class of test configurations.

(Huang): A Kazhdan-Lusztig variety is the intersection of a locally-closed Schubert cell with an opposite Schubert variety in a flag variety. We present a linear parametrization of the Schubert cells in the affine type A flag variety via Bott-Samelson maps, and give explicit equations that generate the Kazhdan-Lusztig ideals in these coordinates. Furthermore, our equations form a Gröbner basis for the Kazhdan-Lusztig ideals. Our result generalizes a result of Woo-Yong that gave a Gröbner basis for Kazhdan-Lusztig ideals in the type A flag variety. This is joint work with Balázs Elek.

(Wang): Let A be an abelian variety defined over a number field K. The well-known Mordell-Weil theorem states that for any finite extension L/K, the torsion subgroup of A(K) is finite. However, over the algebraic closure K^alg, the torsion subgroup of A(K^alg) is infinite. Therefore, a natural question arises: does the finiteness property of the torsion subgroup of A(L) hold for various infinite algebraic extensions L/K? In this talk, we will explore this question in the context where L is the "torsion field" of a different abelian variety. This is joint work with Jeff Achter and Lian Duan. If time permits, we will also formulate this question in a much more general setting and try to ask some new questions.

Past Seminars

Ohio State University Algebraic Geometry Seminar-Year 2021-2022

Ohio State University Algebraic Geometry Seminar-Year 2020-2021

Ohio State University Algebraic Geometry Seminar-Year 2019-2020

Ohio State University Algebraic Geometry Seminar-Year 2018-2019

Ohio State University Algebraic Geometry Seminar-Year 2017-2018

Ohio State University Algebraic Geometry Seminar-Year 2016-2017

Ohio State University Algebraic Geometry Seminar-Year 2015-2016

Ohio State University Algebraic Geometry Seminar-Year 2014-2015

Ohio State University Algebraic Geometry Seminar-Year 2013-2014

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