Cones of effective cycles on blow ups of projective spaces along rational curves

TL;DR

This paper studies the cones of effective cycles on blow ups of projective spaces along rational curves, providing explicit descriptions for certain cases and exploring the geometry of secant varieties.

Contribution

It explicitly determines the cones of divisors and low-dimensional cycles on blow ups along rational normal curves, advancing understanding of their geometric properties.

Findings

Explicit cones of divisors and 1- and 2-cycles for blow ups along rational normal curves

Strengthened results in low-dimensional cases

Computations of effective cycles of secant varieties of blown-up curves

Abstract

In this paper we examine the cones of effective cycles on blow ups of projective spaces along smooth rational curves. We determine explicitly the cones of divisors and 1- and 2-dimensional cycles on blow ups of rational normal curves, and strengthen these results in cases of low dimension. Central to our results is the geometry of resolutions of the secant varieties of the curves which are blown up, and our computations of their effective cycles may be of independent interest.