Determinantal ideals of secant varieties

2025/10/02 by Agostini, Daniele, Park, Jinhyung · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.01895

Abstract

Using Hilbert schemes of points, we establish a number of results for a smooth projective variety X in a sufficiently ample embedding. If X is a curve or a surface, we show that the ideals of higher secant varieties are determinantally presented, and we prove the same for the first secant variety if X has arbitrary dimension. This completely settles a conjecture of Eisenbud-Koh-Stillman for curves and partially resolves a conjecture of Sidman-Smith in higher dimensions. If X is a curve or a surface we also prove that the corresponding embedding of the Hilbert scheme of points X^[d] into the Grassmannian is projectively normal. Finally, if X is an arbitrary projective scheme in a sufficiently ample embedding, then we demonstrate that its homogeneous ideal is generated by quadrics of rank three, confirming a conjecture of Han-Lee-Moon-Park. Along the way, we check that the Hilbert scheme of three points on a smooth variety is the blow-up of the symmetric product along the big diagonal.

Citations

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Cited by

• Quadratic equations of tangent varieties via four-way tensors of linear forms

Related

• Singularities and syzygies of secant varieties of smooth projective varieties
• Ulrich sheaves and determinantal representations for higher secant varieties of curves
• A few remarks about the Hilbert scheme of smooth projective curves
• The Degree of the Tangent and Secant Variety to a Projective Surface
• On the dimension of the Hilbert scheme of curves