Toric fiber products versus Segre products

TL;DR

This paper compares toric fiber products and Segre products, showing how they relate and differ, and explores conditions under which they coincide or have desirable properties in algebraic geometry.

Contribution

It demonstrates that any Segre product can be represented as a toric fiber product, enabling the transfer of results and techniques between these constructions.

Findings

Segre product can be presented as a toric fiber product

Criteria for density and finiteness of the Segre in the toric fiber product

Examples showing non-normality of certain toric fiber products

Abstract

The toric fiber product is an operation that combines two ideals that are homogeneous with respect to a grading by an affine monoid. The Segre product is a related construction that combines two multigraded rings. The quotient ring by a toric fiber product of two ideals is a subring of the Segre product, but in general this inclusion is strict. We contrast the two constructions and show that any Segre product can be presented as a toric fiber product without changing the involved quotient rings. This allows to apply previous results about toric fiber products to the study of Segre products. We give criteria for the Segre product of two affine toric varieties to be dense in their toric fiber product, and for the map from the Segre product to the toric fiber product to be finite. We give an example that shows that the quotient ring of a toric fiber product of normal ideals need not be normal. In rings with Veronese type gradings, we find examples of toric fiber products that are always Segre products, and we show that iterated toric fiber products of Veronese ideals over Veronese rings are normal.