arXiv subjects

Benjamin Gould

Publications and source records attributed to Benjamin Gould.

4 recordsLinked to original sources

arXiv2024

On decompositions for Fano schemes of intersections of two quadrics

We propose conjectural semiorthogonal decompositions for Fano schemes of linear subspaces on intersections of two quadrics, in terms of symmetric powers of the associated hyperelliptic (resp. stacky) curve. When the intersection is odd-dimensional, we moreover conjecture an identity in the Grothendieck ring of varieties and other motivic contexts. The evidence for these conjectures is given by upgrading recent results of Chen-Vilonen-Xue, to obtain formulae for the Hodge numbers of these Fano schemes. This allows us to numerically verify the conjecture in the hyperelliptic case, and establish a combinatorial identity as evidence for the stacky case.

math.AG↗

arXiv2023

Cones of effective cycles on blow ups of projective spaces along rational curves

In this paper we examine the cones of effective cycles on blow ups of projective spaces along smooth rational curves. We determine explicitly the cones of divisors and 1- and 2-dimensional cycles on blow ups of rational normal curves, and strengthen these results in cases of low dimension. Central to our results is the geometry of resolutions of the secant varieties of the curves which are blown up, and our computations of their effective cycles may be of independent interest.

math.AG↗

arXiv2022

Constructive exceptional bundles on $\mathbb{P}^3$

We give a complete classification of the Chern characters of constructive exceptional vector bundles on $\mathbb{P}^3$ analogous to the work of Dr\'ezet and Le Potier on $\mathbb{P}^2$, and using this classification prove that a constructive exceptional bundle $E$ on $\mathbb{P}^3$ with $\mu(E) \geq 0$ is globally generated.

math.AG↗

arXiv2022

Higher rank Brill-Noether theory on P^2

Let $M_{\mathbb{P}^2}(v)$ be a moduli space of semistable sheaves on $\mathbb{P}^2$, and let $B^k(v) \subseteq M_{\mathbb{P}^2}(v)$ be the \textit{Brill-Noether locus} of sheaves $E$ with $h^0(\mathbb{P}^2, E) \geq k$. In this paper we develop the foundational properties of Brill-Noether loci on $\mathbb{P}^2$. Set $r = r(E)$ to be the rank and $c_1, c_2$ the Chern classes. The Brill-Noether loci have natural determinantal scheme structures and expected dimensions $dim B^k(v) = dim M_{\mathbb{P}^2}(v) - k(k - \chi(E))$. When $c_1 > 0$, we show that the Brill-Noether locus $B^r(v)$ is nonempty. When $c_1 = 1$, we show all of the Brill-Noether loci are irreducible and of the expected dimension. We show that when $\mu = c_1/r > 1/2$ is not an integer and $c_2 \gg 0$, the Brill-Noether loci are reducible and describe distinct irreducible components of both expected and unexpected dimension.

math.AG↗