On the effective cone of $\mathbb{P}^n$ blown-up at $n+3$ points

TL;DR

This paper characterizes the effective and movable cones of divisors on the blow-up of projective space at n+3 points, revealing the structure of base loci and proposing a conjecture on linear systems' speciality.

Contribution

It computes the facets of cones of divisors on the blow-up of P^n at n+3 points and introduces a new conjecture relating base locus subvarieties to linear speciality.

Findings

Identified the cycles forming the base locus of linear systems

Computed multiplicities of secant varieties and joins

Proposed a new formula for expected dimension

Abstract

We compute the facets of the effective and movable cones of divisors on the blow-up of $\mathbb{P}^n$ at $n+3$ points in general position. Given any linear system of hypersurfaces of $\mathbb{P}^n$ based at $n+3$ multiple points in general position, we prove that the secant varieties to the rational normal curve of degree $n$ passing through the points, as well as their joins with linear subspaces spanned by some of the points, are cycles of the base locus and we compute their multiplicity. We conjecture that a linear system with $n+3$ points is linearly special only if it contains such subvarieties in the base locus and we give a new formula for the expected dimension.