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Artie Prendergast-Smith

Publications and source records attributed to Artie Prendergast-Smith.

10 recordsLinked to original sources
arXiv2026

Bilinear secants and birational geometry of blowups of $\mathbb P^n \times \mathbb P^{n+1}$

We introduce bilinear secant varieties and joins of subvarieties of products of projective spaces, as a generalisation of the classical secant varieties and joins of projective varieties. We show that the bilinear secant varieties of certain rational normal curves of $\mathbb P^n \times \mathbb P^{n+1}$ play a central role in the study of the birational geometry of $X^{n,n+1}_s$, its blowup in $s$ points in general position. We show that $X^{n,n+1}_s$ is log Fano, and we compute its effective and movable cones, for $s\le n+2$ and $n\ge 1$ and for $s\le n+3$ and $n\le 2$, and we compute the effective and movable cones of $X^{3,4}_6$.

math.AG↗
arXiv2024

Remark on a theorem of Oguiso

For a Calabi-Yau variety X, Oguiso gave a useful criterion for primitivity of a self-map of X in terms of the associated linear map on the Neron--Severi space of X. In this short note, we prove a variant of Oguiso's criterion and use it to verify primitivity of a certain birational automorphism of a Calabi--Yau threefold, to which Oguiso's original criterion does not apply.

math.AG↗
arXiv2015

Fano manifolds of index n-2 and the cone conjecture

The Morrison--Kawamata Cone Conjecture predicts that the action of the automorphism group on the effective nef cone and the action of the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair have rational, polyhedral fundamental domains. In a previous paper, we proved the conjecture for certain blowups of Fano manifolds of index n-1. In this paper, we consider the Morrison--Kawamata conjecture for blowups of Fano manifolds of index n-2.

math.AG↗
arXiv2014

Partially ample line bundles on toric varieties

In this note we study properties of partially ample line bundles on simplicial projective toric varieties. We prove that the cone of q-ample line bundles is a union of rational polyhedral cones, and calculate these cones in examples. We prove a restriction theorem for big q-ample line bundles, and deduce that q-ampleness of the anticanonical bundle is not invariant under flips. Finally we prove a Kodaira-type vanishing theorem for q-ample line bundles.

math.AG↗
arXiv2012

Fano manifolds of index n-1 and the cone conjecture

The Morrison-Kawamata cone conjecture predicts that the actions of the automorphism group on the effective nef cone and the pseudo-automorphism group on the effective movable cone of a klt Calabi-Yau pair $(X, Δ)$ have finite, rational polyhedral fundamental domains. Let $Z$ be an $n$-dimensional Fano manifold of index $n-1$ such that $-K_Z = (n-1) H$ for an ample divisor $H$. Let $Γ$ be the base locus of a general $(n-1)$-dimensional linear system $V \subset |H|$. In this paper, we verify the Morrison-Kawamata cone conjecture for the blow-up of $Z$ along $Γ$.

math.AG↗
arXiv2010

The cone conjecture for abelian varieties

We give a proof of the Morrison--Kawamata cone conjecture for abelian varieties. The proof is a straightforward deduction from well-known results on the real endomorphism algebra of an abelian variety and reduction theory for self-dual homogeneous convex cones.

math.AG↗
arXiv2009

The cone conjecture for some rational elliptic threefolds

We verify the Morrison--Kawamata conjecture for a certain class of rational threefolds, namely blowups of P^3 in the base locus of a net of quadrics with no reducible members. This seems to be the first verified case of the conjecture for klt Calabi--Yau pairs in dimension 3 with non-zero boundary divisor.

math.AG↗