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

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.

22 pages. Revised version. Statement of Theorem 5.1 corrected. To appear in Experimental Mathematics

References in corpus (2)

• Vanishing theorems for linearly obstructed divisors
• On linear systems of ${\mathbb{P}}^3$ with nine base points

Cited by in corpus (4)

• Explicit log Fano structures on blow-ups of projective spaces
• Positivity of divisors on blown-up projective spaces, II
• On linear systems of ${\mathbb{P}}^3$ with nine base points
• Cones of effective divisors on the blown-up P^3 in general lines