Birational transformations on irreducible compact Hermitian symmetric spaces

TL;DR

This paper constructs explicit birational transformations on irreducible compact Hermitian symmetric spaces, transforming them into projective spaces and resolving a known birational map, using loci of minimal rational curves and secant varieties.

Contribution

It provides explicit blow-up and blow-down sequences that convert Hermitian symmetric spaces into projective spaces, resolving a previously known birational map.

Findings

Explicit sequences of blow-ups and blow-downs constructed

Resolution of Landsberg and Manivel's birational map achieved

Centers of blow-ups characterized by minimal rational curves and secant varieties

Abstract

We construct a sequence of explicit blow-ups and blow-downs on irreducible compact Hermitian symmetric spaces $X$ which transforms it into a projective space of the same dimension. Moreover this resolves a birational map given by Landsberg and Manivel. Centers of the blow-ups for $X$ are constructed by loci of chains of minimal rational curves and centers of the blow-ups for the projective space are constructed from the variety of minimal rational tangents of $X$ and its higher secant varieties.