Dennis Tseng's research while affiliated with Devry College of New York, USA and other places
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Publications (22)
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.
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 Mg¯. We determine explicitly...
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.
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 Ch...
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-algebra...
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, w...
We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field k k , this enrichment counts the number of lines meeting four lines defined over k k in P k 3 \mathbf {P}^3_k , with such lines weighted by their fields of definition together with informa...
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...
For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain tautological classes in the combinatorial Chow ring $A^\bullet(M)$ arising from Grassmannians. Using mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the h-vector of a matroid complex, improving on an old c...
We compute the GLr+1-equivariant Chow class of the GLr+1-orbit closure of any point (x1,…,xn)∈(Pr)n in terms of the rank polytope of the matroid represented by x1,…,xn∈Pr. Using these classes and generalizations involving point configurations in higher dimensional projective spaces, we define for each d×n matrix M an n-ary operation [M]ħ on the sma...
The notion of $\mathbb{A}^1$-degree provides an arithmetic refinement of the usual notion of degree in algebraic geometry. In this note, we compute $\mathbb{A}^1$-degrees 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 varietie...
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 on...
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.
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 topologi...
In this note, we extend work of Farkas and Rim\'anyi 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 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 aris...
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 de...
We prove a sharp bound for the dimension of a complete family of smooth rational curves immersed into projective space.
We show the space of smooth 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 irreducible and of the expected dimension. This proves more cases of a conjecture of Coskun, Harris, and Starr.
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 o...
Citations
... Moreover, we know from [ST21] that the pullback homomorphism CH * (B PGL 2 ) → CH * ([PV k / PGL 2 ]) sends c 2 → −(u − v) 2 and c 3 → 0. This implies that ...
... This ring is one with highly arithmetic properties, with elements given by formal differences of quadratic forms of the base field, see definition 2.1. The Grothendieck-Witt ring is comes equipped with a homomorphism, rank : W(k) → Z, and the element we obtain is an "arithmetic refinement", in the sense that if we apply the rank homomorphism, we recover the integer obtained by the classical count over C. Examples of this are an arithmetic refinement of the count of "27 lines on a smooth cubic surface" in [20], and an arithmetic refinement of the count of "lines meeting 4 lines in P 3 " in [40]. ...
Reference: An arithmetic Yau-Zaslow formula
... An algebraic proof of the product rule was given by [5], which we revisit with a few minor corrections. We also obtain a complete classification of the quadratic forms that are realizable as local degrees over finite fields of odd characteristic. ...
... Weighted orbit class in the Schubert basis 2, 2, 2, 2, 2, 2 24 · Ω 3 + 48 · Ω 2,1 3, 2, 2, 2, 2 16 · Ω 3 + 40 · Ω 2,1 4, 2, 2, 2 12 · Ω 3 + 24 · Ω 2,1 3, 3, 2, 2 8 · Ω 3 + 32 · Ω 2,1 3, 3, 3 24 · Ω 2,1 4, 3, 2 4 · Ω 3 + 16 · Ω 2,1 Table 1. The weighted orbit classes of base-point free pencils of quartics in Gr(2, Sym 4 k 2 ) as a function of the ramification profile. ...
... Since the seminal work of Mori and Miyaoka [3,4], rational curves on algebraic varieties, especially Fano manifolds, have been much studied. In particular Harris and his school (see for instance [2,8,9] and references therein) have studied the case of rational curves on general Fano hypersurfaces, with particular attention to the question of dimension and irreducibility of the family of curves of given degree. ...
Reference: Incident rational curves
... ∈ P(Sym d (W )), dim V(h) sing ≥ 1}. This space has been studied in [Sla15;Tse20]. In the case of cubic surfaces, [Suk20] gives a classification of the cubic surfaces with positive-dimensional singular locus: it consists of the reducible cubics (which forms a variety of dimension 12) and of the orbit closure under the PGL 4 -action of an irreducible cubic surface corresponding to [Suk20, Table 1, 6C], giving a variety of dimension 13. ...
