Fano manifolds and blow-ups of low-dimensional subvarieties

TL;DR

This paper classifies certain high-dimensional Fano manifolds obtained by blowing up smooth varieties along subvarieties of specific dimensions, revealing their cone structures and unique contraction properties.

Contribution

It provides a classification of the cones of curves for these Fano manifolds and identifies the unique manifold lacking a fiber type elementary contraction.

Findings

Classification of cones of curves for these Fano manifolds

Identification of the unique manifold without fiber type contraction

Structural insights into blow-ups along subvarieties of dimension equal to the pseudoindex

Abstract

We study Fano manifolds of pseudoindex greater than one and dimension greater than five, which are blow-ups of smooth varieties along smooth centers of dimension equal to the pseudoindex of the manifold. We obtain a classification of the possible cones of curves of these manifolds, and we prove that there is only one such manifold without a fiber type elementary contraction.