Determinantal ideals of secant varieties

TL;DR

The paper develops a unified framework for when secant-variety ideals $I(\\Sigma_k(X,L))$ are determinantally presented by $(k+2)\\times(k+2)$ minors of a catalecticant-like matrix $\\operatorname{Cat}(A,B)$, using tautological bundles on Hilbert schemes $X^{[k+2]}$ and two complementary proof strategies: a Hilbert-Chow–based reduction and a cohomological vanishings approach. It proves determinantally presented ideals for curves (and, with mild assumptions, for higher dimensions in the first secant case) and extends to surfaces; it also shows projective normality of Hilbert-scheme embeddings into Grassmannians for curves and surfaces, and establishes that $I(X,L)$ is generated by quadrics of rank $3$ for any projective scheme in sufficiently ample embedding, resolving conjectures of Eisenbud–Koh–Stillman, Sidman–Smith, and Han–Lee–Moon–Park in the stated regimes. A key methodological theme is translating multiplication maps on global sections into geometry on $X^{[k+2]}$, enabling effective, even quantitative, criteria (e.g., degree bounds) for determinantally presented ideals via tautological bundles and cohomology vanishing. The results yield explicit, computationally accessible equations for secant varieties, sharpen degrees needed for determinant presentations, and a clear link between Hilbert-scheme positivity and the algebraic structure of secant-variety ideals. The work also clarifies the Hilbert-Chow morphism as a blow-up in the relevant cases and provides a robust pathway to study syzygies and normality in related moduli spaces.

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.

Theorems & Definitions (63)

• Theorem A
• Theorem B
• Theorem C
• Proposition D
• Theorem E
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3