Blowups and fibers of morphisms

TL;DR

This paper investigates the properties of rational maps between projective spaces, relating algebraic conditions of associated rings to geometric features like birationality, and provides criteria and methods to analyze and reduce non-birational cases.

Contribution

It establishes algebraic conditions characterizing birationality of rational maps via properties of Rees algebras and introduces explicit methods to handle non-birational cases when s=2.

Findings

R[It] satisfies Serre's R_i condition iff A satisfies R_{i-1} and Psi is birational.

Psi is birational iff R[It] satisfies R_1.

Explicit reduction method for non-birational to birational cases when s=2.

Abstract

Our object of study is a rational map Psi from projective s-1 space to projective n-1 space defined by homogeneous forms g1,...,gn, of the same degree d, in the homogeneous coordinate ring R=k[x1,...,xs] of projective s-1 space. Our goal is to relate properties of Psi, of the homogeneous coordinate ring A=k[g1,...,gn] of the variety parametrized by Psi, and of the Rees algebra R[It], the bihomogeneous coordinate ring of the graph of Psi. For a regular map Psi, for instance, we prove that R[It] satisfies Serre's condition R_i, for some positive i, if and only if A satisfies R_{i-1} and Psi is birational onto its image. Thus, in particular, Psi is birational onto its image if and only if R[It] satisfies R_1. Either condition has implications for the shape of the core, namely, the core of I is the multiplier ideal of I to the power s and the core of I equals the maximal homogeneous ideal of R to the power sd-s+1. Conversely, for s equal to two, either equality for the core implies birationality. In addition, by means of the generalized rows of the syzygy matrix of g1,...,gn, we give an explicit method to reduce the non-birational case to the birational one when s is equal to 2.