Mapping toric varieties into low dimensional spaces

TL;DR

This paper investigates the minimal embedding dimension for injective maps of normal toric varieties, especially Segre-Veronese varieties, into projective spaces, highlighting differences between smooth and singular cases.

Contribution

It provides new bounds and results on the minimal dimension needed for injective embeddings of normal toric varieties, focusing on Segre-Veronese varieties.

Findings

Normal toric varieties can be embedded injectively into low-dimensional projective spaces.

Complete results are obtained for Segre-Veronese varieties.

The minimal embedding dimension depends on the variety's singularities and structure.

Abstract

A smooth $d$-dimensional projective variety $X$ can always be embedded into $2d+1$-dimensional space. In contrast, a singular variety may require an arbitrary large ambient space. If we relax our requirement and ask only that the map is injective, then any $d$-dimensional projective variety can be mapped injectively to $2d+1$-dimensional projective space. A natural question then arises: what is the minimal $m$ such that a projective variety can be mapped injectively to $m$-dimensional projective space? In this paper we investigate this question for normal toric varieties, with our most complete results being for Segre-Veronese varieties.