A note on the very ampleness of complete linear systems on blowings-up of P^3

TL;DR

This paper establishes criteria for the very ampleness and related properties of linear systems on blow-ups of P^3 at general points on a specific divisor, extending understanding of embedding properties in algebraic geometry.

Contribution

It provides necessary and sufficient conditions for very ampleness of linear systems on certain blow-ups of P^3, including a new sufficient condition for very ampleness.

Findings

Criteria for very ampleness of linear systems on blow-ups of P^3.

Necessary and sufficient conditions for base point freeness.

A new sufficient condition for very ampleness in this setting.

Abstract

In this note we consider the blowing-up X of P^3 along r general points of the anticanonical divisor of a smooth quadric in P^3. Given a complete linear system |L| = |dH - m_1 E_1 -...- m_r E_r| on X, with H the pull-back of a plane in P^3 and E_i the exceptional divisor corresponding to P_i, we give necessary and sufficient conditions for the very ampleness (resp. base point freeness and non-speciality) of L. As a corollary we obtain a sufficient condition for the very ampleness of such a complete linear system on the blowing-up of P^3 along r general points.