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Effective cone of a Grassmann bundle over a curve defined over $\overline{\mathbb F}_p$

Indranil Biswas, Shripad M. Garge, Krishna Hanumanthu

Abstract

Let $X$ be an irreducible smooth projective curve defined over $\overline{\mathbb F}_p$ and $E$ a vector bundle on $X$ of rank at least two. For any $1\, \leq\, r\, <\, {\rm rank}(E)$, let ${\rm Gr}_r(E)$ be the Grassmann bundle over $X$ parametrizing all the $r$ dimensional quotients of the fibers of $E$. We prove that the effective cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$ coincides with the pseudo-effective cone in ${\rm NS}({\rm Gr}_r(E))\otimes_{\mathbb Z} {\mathbb R}$. When $r\,=\,1$ or ${\rm rank}(E)-1$, this was proved by A. Moriwaki.

Effective cone of a Grassmann bundle over a curve defined over $\overline{\mathbb F}_p$

Abstract

Let be an irreducible smooth projective curve defined over and a vector bundle on of rank at least two. For any , let be the Grassmann bundle over parametrizing all the dimensional quotients of the fibers of . We prove that the effective cone in coincides with the pseudo-effective cone in . When or , this was proved by A. Moriwaki.
Paper Structure (5 sections, 3 theorems, 25 equations)

This paper contains 5 sections, 3 theorems, 25 equations.

Table of Contents

  1. introduction
  2. Pseudo-effective cone of a Grassmann bundle
  3. When the characteristic is zero
  4. When the characteristic is positive
  5. The effective cone

Key Result

Lemma 2.1

The pseudo-effective cone of ${\rm Gr}_r(E)$ is generated by $\phi^*c_1(L)$ and $c_1({\mathcal{O}}_{{\rm Gr}_r(E)}(1))- \lambda \phi^*c_1(L)$, where $\lambda$ is defined in e4.

Theorems & Definitions (4)

  • Lemma 2.1: BHP
  • Lemma 2.2: BHP
  • Theorem 3.1
  • proof