Singularities and syzygies of secant varieties of smooth projective varieties

TL;DR

This paper investigates the geometric and algebraic properties of secant varieties of smooth projective varieties, establishing conditions for normality, singularity types, and syzygy behaviors, with results extending to higher dimensions and revealing limitations.

Contribution

It provides new results on the singularities and syzygies of secant varieties, especially for surfaces, and characterizes their ideals via symbolic powers, also exploring higher-dimensional cases.

Findings

Secant varieties of surfaces are normal with Du Bois singularities under positive embeddings.

Syzygies of defining ideals are linear up to the expected order for certain secant varieties.

The cohomology of the surface determines the Cohen--Macaulay and rational singularity properties of its secant varieties.

Abstract

We study the higher secant varieties of a smooth projective variety embedded in projective space. We prove that when the variety is a surface and the embedding line bundle is sufficiently positive, these varieties are normal with Du Bois singularities and the syzygies of their defining ideals are linear to the expected order. We show that the cohomology of the structure sheaf of the surface completely determines whether the singularities of its secant varieties are Cohen--Macaulay or rational. We also prove analogous results when the dimension of the original variety is higher and the secant order is low, and by contrast we prove a result that strongly implies these statements do not generalize to higher dimensional varieties when the secant order is high. Finally, we deduce a complementary result characterizing the ideal of secant varieties of a surface in terms of the symbolic powers of the ideal of the surface itself, and we include a theorem concerning the weight one syzygies of an embedded surface -- analogous to the gonality conjecture for curves -- which we discovered as a natural application of our techniques.