Poster abstracts
WAGS UC Riverside Fall 2022
Mont Cordero Aguilar
The tropical degree of a tropical root surface
The field of tropical geometry arose from the desire to convert an algebraic variety V into a piecewise linear combinatorial structure Trop V that retains a lot of information about V, such as degree, dimension, etc. We study the tropical surfaces that arise from the root systems of type A rather than from the tropicalization of an algebraic variety. Our main result is that the tropical root surface of A_{n-1} has degree (1/2)(n^3-3n^2-2n+12).
Minimal exponent of local complete intersection varieties
The minimal exponent for hypersurfaces is a singularity invariant which refines the log canonical threshold, due to Saito. It was studied in the 1980s in the case of isolated singularities through the Steenbrink Spectrum, and it has recently been related to Hodge ideals. In joint work with Chen, Mustaţă and Olano, we define the invariant for LCI varieties. I will explain the definition, list properties and examples, and explain the connection to Hodge filtration on local cohomology.
Quotient singularities in the Grothendieck ring of varieties
The class of a smooth projective variety in the Grothendieck ring of varieties contains a great deal of geometric information. For example, a famous result of Larsen and Lunts shows that this ring detects stable rationality of varieties that are smooth and projective. The interpretation of classes of singular varieties is less clear, but for a certain class of singular varieties, we can "pretend" they are smooth for the purpose of detecting stable rationality. In joint work with Federico Scavia (https://arxiv.org/abs/2208.14313), we explore the question of which quotient varieties belong to this class.
The hidden structure of negative curves
The problem of determining the MDS property for blowups of weighted projective planes (WPP) at a general point has received renewed interest because of the essential role it plays in Castravet and Tevelev’s proof that M_{0,n} is not a MDS for n>>1. Such a blowup is a MDS if and only if it contains a non-exceptional negative curve and another curve disjoint from it. From a toric perspective, a WPP is defined by a rational plane triangle. We consider a parameter space of triangles and see how the negative curves and the MDS property vary within it. Using this approach we recover and expand most known results in the area, including examples that do not contain a non-exceptional negative curve. This is joint work with Jose Luis González and Kalle Karu.
The $F$-signature function on the ample cone
F-signature plays a crucial role when measuring singularities of varieties in positive characteristics. For example, if R is a local ring, s(R) = 1 implies R is regular, and 0<s(R)<1 implies R is strongly F-regular, which is a char p analog of klt singularities. For a globally F-regular variety X, we define the global F-signature as the F-signature of the section along an invertible sheaf L over X. This poster presents the global F-signature as well-defined and continuous on the ample cone and has the continuous extension to the boundary of the cone. This is joint work with Swaraj Pande.
Yuze Luan
Irreducible components of Hilbert scheme of points on non-reduced curves
We classify the irreducible components of the Hilbert scheme of $n$ points on non-reduced algebraic plane curves. The irreducible components are indexed by partitions and all have dimension $n$.
Normal bundles of rational normal curves on hypersurfaces
Let $C$ be the rational normal curve of degree $e$ in $\mathbb{P}^n$, and let $X\subset \mathbb{P}^n$ be a degree $d\ge 2$ hypersurface containing $C$. I. Coskun and E. Riedl showed that the normal bundle $N_{C/X}$ is balanced for a general $X$. H. Larson studied the case of lines ($e=1$) and computed the dimension of the space of hypersurfaces for which $N_{C/X}$ has a given splitting type. This poster is based on a paper where we work with any $e\ge 2$. We compute explicit examples of hypersurfaces for all possible splitting types, and for $d\ge 3$, we compute the dimension of the space of hypersurfaces for which $N_{C/X}$ has a given splitting type. For $d=2$, we give a lower bound on the maximum rank of quadrics with fixed splitting type.
Construction of varieties of low codimension with applications to moduli spaces of varieties of general type
In this article we develop a new way of systematically constructing infinitely many families of smooth subvarieties $X$ of any given dimension $m$, $m \geq 3$, and any given codimension in $\mathbb {P}^N$, embedded by complete subcanonical linear series, and, in particular, in the range of Hartshorne's conjecture. We accomplish this by showing the existence of everywhere non--reduced schemes called ropes, embedded in {{$\mathbb P^N$}}, and by smoothing them. In the range $3 \leq m \leq {{N/2}}$, we construct smooth subvarieties, embedded by complete subcanonical linear series, that are not complete intersections. We also go beyond a question of Enriques on constructing simple canonical surfaces in projective spaces, and construct simple canonical varieties in all dimensions. The canonical map of infinitely many of these simple canonical varieties is finite birational but not an embedding. Finally, we show the existence of components of moduli spaces of varieties of general type (in all dimensions $m$, $m \geq 3$) that are analogues of the moduli space of curves of genus $g > 2$ with respect to the behavior of the canonical map and its deformations. In many cases, the general elements of these components are canonically embedded and their codimension is in the range of Hartshorne's conjecture.
Lauren Nowak and Patrick O'Melveny
Mixed volumes of normal complexes
In 2021, Nathanson and Ross demonstrated that a geometric object called a normal complex is the correct object to unite the algebraic concept of a tropical fan's volume polynomial with an actual geometric volume computation. That same year, expanding on the work of Adiprasito, Huh, and Katz in proving log-concavity in characteristic polynomials of matroids, Amini and Piquerez established that mixed degrees of divisors of certain classes of tropical fans are log-concave. Given that mixed volumes also generate log-concave sequences, we develop a definition of mixed volumes of normal complexes and establish log-concavity for mixed degrees of divisors of a broader class of tropical fans.
The geometry of sextics and resolvent problems
Here’s one of the fundamental problems of mathematics: when nature hands you a single variable polynomial, determine its roots in the simplest manner possible. As it turns out, we’re really bad at solving this problem. Simple solutions to algebraic equations are often obstructed due to the underlying geometry of these problems. In a very precise sense, via the language of resolvent degree, we don’t even know if sextics are more complicated to solve than quintics. We will set the stage to discuss resolvent problems and analytic solutions to polynomials arising from uniformizations, and towards the end touch upon original work that indicates sextics are indeed qualitatively more complex than quintics.
A Horrocks-type splitting criterion for smooth projective toric varieties of Picard rank 2
In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Motivated by the study of indecomposable vector bundles, in 1978 Beilinson constructed a resolution of the diagonal on P^n which has been used to great effect in algebraic geometry. We obtain a Horrocks-type splitting criterion (under an additional hypothesis) for vector bundles over a smooth projective toric variety X of Picard rank 2 using a linear resolution of the diagonal consisting of finite direct sums of line bundles. Since this resolution has length dim(X), we also prove a new case of a conjecture of Berkesch--Erman--Smith that predicts a version of Hilbert's Syzygy Theorem for virtual resolutions. This is joint work with Michael Brown.
Degenerate fourfold Massey products over arbitrary fields
We prove that, for all fields F of characteristic different from 2 and all a,b,c in F*, the mod 2 Massey product <a,b,c,a> vanishes as soon as it is defined. For every field E of characteristic different form 2, we construct a field F containing E and a,b,c,d in F* such that <a,b,c> and <b,c,d> vanish but <a,b,c,d> is not defined. As a consequence, we answer a question of Positselski by constructing the first examples of fields containing all roots of unity and such that the mod 2 cochain DGA of the absolute Galois group is not formal. This is joint work with Alexander Merkurjev.
Tonie Scroggin
Computing the homology of braid varieties on two strands using cluster algebras
In 1923, Alexander proved that every knot/link can be represented as the closure of a braid. Similar to the Reidemeister moves on knots, any two closures represent the same link if and only if the braids are related by a sequence of stabilizations and conjugations, which are referred to as Markov moves. This allows one to construct link invariants by assigning objects to the crossings $\sigma_i^\pm$, verifying that braid relations are satisfied, defining a closure operation and checking that the result is invariant under Markov moves. We define the braid variety. The homology of a braid variety is related to the Khovanov-Rozansky homology of a corresponding link. The braid variety is isomorphic to a positroid variety; therefore, the braid variety has a cluster structure. Any cluster structure has a canonical 2-form with constant coefficients in all cluster charts, which gives an interesting class in de Rham cohomology of degree 2, and an interesting operator in link homology. Using the cluster structure, we compute the homology of the braid variety using the De Rham cohomology.
Tautological bundles over Quot scheme of curves
We find explicit formulas for the Euler characteristics of tautological bundles over punctual Quot schemes of smooth projective curve C that parameterize zero-dimensional quotients of a vector bundle E over C. The formulas suggest analogies between the Quot schemes of curves and the Hilbert scheme of points of surfaces. Our proofs rely on Atiyah-Bott localization, universality results (of Ellingsrud, Gottsche, and Lehn), and the combinatorics of Schur functions. For higher rank quotients, we obtain expressions in genus 0. This is joint work with Dragos Oprea.
Non-algebraic geometrically trivial cohomology classes over finite fields
The Tate conjecture is a central problem in arithmetic geometry, describing algebraic cycles on an algebraic variety in terms of Galois representation on étale cohomology. In contrast, an integral analogue of the Tate conjecture is known to fail in general, and a natural question is whether the failure of the integral Tate conjecture is due to geometric reasons. Over a finite field, we construct the first counterexamples to this question: in codimension 2 on our examples, a geometric cycle map is surjective but an arithmetic cycle map is not. We also show positive results toward a conjecture of Colliot-Thélène and Kahn on the third unramified cohomology group for threefolds over a finite field. This is a joint work with Federico Scavia.
On the anisotropy theorem of Papadakis and Petrotou
Papadakis and Petrotou showed that anisotropy of a quadratic form implies the Hard Lefschetz theorem for simplicial spheres in characteristic 2. We use an integration map to define a mixed volume on spheres, which is linear over a connected sum decomposition. Using this decomposition, we explicitly describe the quadratic form and prove an additional combinatorial conjecture of Papadakis and Petrotou, from which all characteristic 2 anisotropy theorems follow. This is joint work with Kalle Karu.
Lingyao Xie
Semi-ampleness of generalized pairs
Generalized pairs (g-pair for short) naturally appear in canonical bundle formulas and are important in solving BAB conjecture. We generalized many results in Minimal Model Program of log canonical pairs to the setting of generalized pairs, and especially focus on the semi-ampleness part, which turn out to be really subtle due to the failure of finiteness of B-representation.
Application of Coxeter groups to the Kawamata - Morrison conjecture
The Kawamata - Morrison conjecture predicts that the convex geometry of the Nef cone and the Movable cone of a Calabi - Yau variety X is connected to the action of its group of automorphism and birational automorphism respectively. From the birational geometry point of view, this implies that the birational contractions and fiber space structures are finite up to automorphism, and that X has finitely many minimal models. We use the theory of Coxeter groups to show the conjecture for complete intersections in products of projective spaces, along with computing explicitly their birational automorphism group and their movable cone. Thi work is joint with Michael Hoff and Isabel Stenger.