On intrinsic quadrics

TL;DR

This paper classifies smooth intrinsic quadrics with small Picard numbers, identifies Fano examples among them, and verifies Fujita's base point free conjecture in this context.

Contribution

It provides explicit descriptions of smooth intrinsic quadrics with small Picard numbers and confirms Fujita's conjecture for these varieties.

Findings

Explicit classifications of smooth intrinsic quadrics for small Picard numbers

Identification of Fano examples within these quadrics

Verification of Fujita's base point free conjecture in this setting

Abstract

An intrinsic quadric is a normal projective variety with a Cox ring defined by a single quadratic relation. We provide explicit descriptions of these varieties in the smooth case for small Picard numbers. As applications, we figure out in this setting the Fano examples and (affirmatively) test Fujita's base point free conjecture.