On secant loci and simple linear projections of some projective varieties

TL;DR

This paper investigates how simple linear projections of projective varieties behave when the projection center varies, revealing that secant loci are either empty or quadrics, which determines the properties of the projected varieties.

Contribution

It establishes a link between secant loci and the properties of projected varieties, providing classifications for Veronese and Segre embeddings and analyzing cohomological properties.

Findings

Secant locus is either empty or a quadric in a subspace.

The projection map is birational.

Cohomological properties of projected varieties are determined by secant loci.

Abstract

In this paper, we study how simple linear projections of some projective varieties behave when the projection center runs through the ambient space. More precisely, let $X \subset \P^r$ be a projective variety satisfying Green-Lazarsfeld's property $N_p$ for some $p \geq 2$, $q \in \P^r$ a closed point outside of $X$, and $X_q := \pi_q (X) \subset \P^{r-1}$ the projected image of $X$ from $q$. First, it is shown that the secant locus $\Sigma_q (X)$ of $X$ with respect to $q$, i.e. the set of all points on $X$ spanning secant lines passing through $q$, is either empty or a quadric in a subspace of $\P^r$. This implies that the finite morphism $\pi_q : X \to X_q$ is birational. Our main result is that cohomological and local properties of $X_q$ are precisely determined by $\Sigma_q (X)$. To complete this result, the next step should be to classify all possible secant loci and to decompose the ambient space via the classification of secant loci. We obtain such a decomposition for Veronese embeddings and Segre embeddings. Also as an application of the main result, we study cohomological properties of low degree varieties.