arXiv subjects

Dennis Tseng

Publications and source records attributed to Dennis Tseng.

At least 19 recordsLinked to original sources

arXiv2023

Tautological classes of matroids

We introduce certain torus-equivariant classes on permutohedral varieties which we call "tautological classes of matroids" as a new geometric framework for studying matroids. Using this framework, we unify and extend many recent developments in matroid theory arising from its interaction with algebraic geometry. We achieve this by establishing a Chow-theoretic description and a log-concavity property for a 4-variable transformation of the Tutte polynomial, and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for permutohedral varieties that translates between K-theory and Chow theory.

math.CO↗

arXiv2023

Log-concavity of matroid h-vectors and mixed Eulerian numbers

For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the $h$-vector of a matroid complex, improving on an old conjecture of Dawson.

math.AG↗

arXiv2022

Lines highly tangent to a hypersurface

We study spaces of lines that meet a smooth hypersurface X in P^n to high order. As an application, we give a polynomial upper bound on the number of planes contained in a smooth degree d hypersurface in P^5 and provide a proof of a result of Landsberg without using moving frames.

math.AG↗

arXiv2022

Equivariant Degenerations of Plane Curve Orbits

In a series of papers, Aluffi and Faber computed the degree of the $GL_3$ orbit closure of an arbitrary plane curve. We attempt to generalize this to the equivariant setting by studying how orbits degenerate under some natural specializations, yielding a fairly complete picture in the case of plane quartics.

math.AG↗

arXiv2022

Moduli of linear slices of high degree hypersurfaces

We study the variation of linear sections of hypersurfaces in $\mathbb{P}^n$. We completely classify all plane curves, necessarily singular, whose line sections do not vary maximally in moduli. In higher dimensions, we prove that the family of hyperplane sections of any smooth degree $d$ hypersurface in $\mathbb{P}^n$ vary maximally for $d \geq n+3$. In the process, we generalize the classical Grauert-Mulich theorem about lines in projective space, both to $k$-planes in projective space and to free rational curves on arbitrary varieties.

math.AG↗

arXiv2021

Divisors on the moduli space of curves from divisorial conditions on hypersurfaces

In this note, we extend work of Farkas and Rimányi on applying quadric rank loci to finding divisors of small slope on the moduli space of curves by instead considering all divisorial conditions on the hypersurfaces of a fixed degree containing a projective curve. This gives rise to a large family of virtual divisors on $\overline{\mathcal{M}_g}$. We determine explicitly which of these divisors are candidate counterexamples to the Slope Conjecture. The potential counterexamples exist on $\overline{\mathcal{M}_g}$, where the set of possible values of $g\in \{1,\ldots,N\}$ has density $Ω(\log(N)^{-0.087})$ for $N>>0$. Furthermore, no divisorial condition defined using hypersurfaces of degree greater than 2 give counterexamples to the Slope Conjecture, and every divisor in our family has slope at least $6+\frac{8}{g+1}$.

math.AG↗

arXiv2021

A Universal Formula For Counting Cubic Surfaces

Using equivariant geometry, we find a universal formula that computes the number of times a general cubic surface arises in a family. As applications, we show that the PGL(4) orbit closure of a generic cubic surface has degree 96120, and that a general cubic surface arises 42120 times as a hyperplane section of a general cubic 3-fold.

math.AG↗

arXiv2020

On the EKL-Degree of a Weyl Cover

More than four decades ago, Eisenbud, Khimšiašvili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. Their notion of degree, which we call the EKL-degree, can be thought of as a refinement of the usual notion of local degree in algebraic geometry that works over non-algebraically closed base fields, taking values in the Grothendieck-Witt ring. In this note, we compute the EKL-degree at the origin of certain finite covers $f\colon \mathbb{A}^n\to \mathbb{A}^n$ induced by quotients under actions of Weyl groups. We use knowledge of the cohomology ring of partial flag varieties as a key input in our proofs, and our computations give interesting explicit examples in the field of $\mathbb{A}^1$-enumerative geometry.

math.AG↗

arXiv2020

Critical groups of covering, voltage, and signed graphs

Graph coverings are known to induce surjections of their critical groups. Here we describe the kernels of these morphisms in terms of data parametrizing the covering. Regular coverings are parametrized by voltage graphs, and the above kernel can be identified with a naturally defined voltage graph critical group. For double covers, the voltage graph is a signed graph, and the theory takes a particularly pleasant form, leading also to a theory of double covers of signed graphs.

math.CO↗

arXiv2020

On degenerate sections of vector bundles

We consider the locus of sections of a vector bundle on a projective scheme that vanish in higher dimension than expected. We show that after applying a high enough twist, any maximal component of this locus consists entirely of sections vanishing along a subscheme of minimal degree. In fact, we will give a more refined description of this locus, which will allow us to deduce its limit in the Grothendieck ring of varieties.

math.AG↗

arXiv2019

Orbits in $(\mathbb{P}^r)^n$ and equivariant quantum cohomology

We compute the $GL_{r+1}$-equivariant Chow class of the $GL_{r+1}$-orbit closure of any point $(x_1, \ldots, x_n) \in (\mathbb{P}^r)^n$ in terms of the rank polytope of the matroid represented by $x_1, \ldots, x_n \in \mathbb{P}^r$. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each $d\times n$ matrix $M$ an $n$-ary operation $[M]_\hbar$ on the small equivariant quantum cohomology ring of $\mathbb{P}^r$, which is the $n$-ary quantum product when $M$ is an invertible matrix. We prove that $M \mapsto [M]_\hbar$ is a valuative matroid polytope association. Like the quantum product, these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting. As an application, we compute the number of line sections with given moduli of a general degree $2r+1$ hypersurface in $\mathbb{P}^r$, generalizing the known case of quintic plane curves.

math.AG↗

arXiv2019

A note on $r$ hypersurfaces intersecting in $\mathbb{P}^r$

We consider the locus of $r$-tuples of homogeneous forms of some fixed degree whose common vanishing locus in $\mathbb{P}^r$ is positive dimensional. We show that any component of maximal dimension of that locus either consists of homogeneous forms all vanishing on some line or homogeneous forms where a proper subset fail to intersect properly.

math.AG↗

arXiv2019

On the slope of the moduli space of genus 15 and 16 curves

We revisit the work of Chang and Ran on bounding the slopes of $\overline{\mathscr{M}_{15}}$ and $\overline{\mathscr{M}_{16}}$, correct one of the formulas used at the conclusion of the argument, and recompute the lower bounds on the slopes, yielding $s(\overline{\mathscr{M}_{15}})>6.5$ but not for $\overline{\mathscr{M}_{16}}$. Our contribution only involves plugging in formulas.

math.AG↗

arXiv2019

Incidence strata of affine varieties with complex multiplicities

To each affine variety $X$ and $m_1,\ldots,m_k\in \mathbb{C}$ such that no subset of the $m_i$ add to zero, we construct a variety which for $m_1,\ldots,m_k \in \mathbb{N}$ specializes to the closed $(m_1,\ldots,m_k)$-incidence stratum of $Sym^{m_1+\ldots+m_k}X$. These fit into a finite-type family, which is functorial in $X$, and which is topologically a family of $\mathbb{C}$-weighted configuration spaces. We verify our construction agrees with an analogous construction in the Deligne category $Rep(S_{d})$ for $d \in \mathbb{C}$. We next classify the singularity locus and branching behaviour of colored incidence strata for arbitrary smooth curves. As an application, we negatively answer a question of Farb and Wolfson concerning the existence of an isomorphism between two natural moduli spaces.

math.AG↗

arXiv2019

A note on rational curves on general Fano hypersurfaces

We show the Kontsevich space of rational curves of degree at most roughly $\frac{2-\sqrt{2}}{2}n$ on a general hypersurface $X\subset \mathbb{P}^n$ of degree $n-1$ is equidimensional of expected dimension and has two components: one consisting generically of smooth, embedded rational curves and the other consisting of multiple covers of a line. This proves more cases of a conjecture of Coskun, Harris, and Starr and shows the Gromov-Witten invariants in these cases are enumerative.

math.AG↗

arXiv2019

Complete families of immersed curves

We extend results of Chang and Ran regarding large dimensional families of immersed curves of positive genus in projective space in two directions. In one direction, we prove a sharp bound for the dimension of a complete family of smooth rational curves immersed into projective space, completing the picture in projective space. In another direction, we isolate the necessary positivity condition on the tangent bundle of projective space used to run the argument, which allows us to rule out large dimensional families of immersed curves of positive genus in generalized flag varieties.

math.AG↗

arXiv2018

$PGL_2$-equivariant strata of point configurations in $\mathbb{P}^1$

We compute the integral Chow ring of the quotient stack $[(\mathbb{P}^1)^n/PGL_2]$, which contains $\mathcal{M}_{0,n}$ as a dense open, and determine a natural $\mathbb{Z}$-basis for the Chow ring in terms of certain ordered incidence strata. We further show that all $\mathbb{Z}$-linear relations between the classes of ordered incidence strata arise from an analogue of the WDVV relations in $A^\bullet(\overline{\mathcal{M}}_{0,n})$. Next we compute the classes of unordered incidence strata in the integral Chow ring of the quotient stack $[{\rm Sym}^n\mathbb{P}^1/PGL_2]$ and classify all $\mathbb{Z}$-linear relations between the strata via these analogues of WDVV relations. Finally, we compute the rational Chow rings of the complement of a union of unordered incidence strata.

math.AG↗

arXiv2018

Collections of hypersurfaces containing a curve

We consider the closed locus of $r$-tuples of hypersurfaces in $\mathbb{P}^r$ with positive dimensional intersection, and show in a large range of degrees that its largest component is the locus of $r$-tuples of hypersurfaces whose intersection contains a line. We then apply our methods to obtain new results on the largest components of the locus of hypersurfaces with positive dimensional singular loci and the locus of smooth hypersurfaces with a higher dimensional family of lines through a point than expected.

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