[Paper Review] On the ideals of Secant Varieties to certain rational varieties

Maria Virginia Catalisano, Anthony V. Geramita+1 more|ArXiv.org|Sep 2, 2006

Tensor decomposition and applications11 references4 citations

TL;DR

This paper determines the defining ideals of higher secant varieties for specific rational varieties, including Segre embeddings of products of projective spaces, Segre-Veronese embeddings with two or three factors, and Del Pezzo surfaces. It proves that these ideals are generated by determinantal equations from flattenings of the associated tensors, extending known results for two-factor products to higher-rank tensors and confirming conjectures on secant line and plane varieties.

ABSTRACT

If $\X \subset ¶^n$ is a reduced and irreducible projective variety, it is interesting to find the equations describing the (higher) secant varieties of $\X$. In this paper we find those equations in the following cases: $\X = ¶^{n_1} imes... imes¶^{n_t} imes¶^n$ is the Segre embedding of the product and $n$ is "large" with respect to the $n_i$ (Theorem 2.4); $\X$ is a Segre-Veronese embedding of some products with 2 or three factors; $\X$ is a Del Pezzo surface.

Motivation & Objective

• To determine the defining ideals of higher secant varieties for certain rational varieties, including Segre embeddings of products of projective spaces with more than two factors.
• To extend the known determinantal description of secant variety ideals—previously valid for two-factor products—to cases with three or more factors.
• To verify and generalize the conjecture that secant variety ideals are generated by minors of tensor flattenings, particularly for secant line and plane varieties.
• To analyze the geometric and algebraic properties of these ideals, including Cohen-Macaulayness, resolution, and degree, using known formulas and computational tools.
• To compare secant varieties of Del Pezzo surfaces and related rational surfaces via numerical invariants and ideal structures.

Proposed method

• Uses tensor flattenings of the homogeneous coordinate tensor associated with the Segre embedding to generate candidate equations for secant variety ideals.
• Applies the conjecture from [GSS] that secant variety ideals are generated by (s+1)×(s+1) minors of flattenings, and proves it for specific cases including three-factor products.
• Employs projection techniques and Remark 4.1 (secant variety of a projection is the projection of the secant variety) to relate ideals of projected varieties to their originals.
• Utilizes computational algebra systems like CoCoA to verify ideal generation, Betti numbers, and degrees in specific examples.
• Applies known results on Hilbert series and Giambelli’s formula for degrees of determinantal varieties to compute invariants.
• Relies on the Eagon-Northcott complex to describe minimal free resolutions of determinantal ideals, especially for generic height cases.

Experimental results

Research questions

• RQ1 Are the defining ideals of higher secant varieties of Segre embeddings of t-factor products (t ≥ 3) generated by the (s+1)×(s+1) minors of all possible flattenings of the associated tensor?
• RQ2 Do the secant line and plane varieties of Segre-Veronese embeddings of products with two or three factors have ideals generated by determinantal equations from flattenings?
• RQ3 Is the ideal of the secant variety of a Del Pezzo surface generated by 3×3 minors of a specific matrix derived from the embedding?
• RQ4 For projected rational surfaces like S8 and D8, do the secant variety ideals coincide with those generated by 3×3 minors of a matrix, despite differing geometric behavior in higher secants?
• RQ5 Can numerical invariants like degree and Betti numbers distinguish between secant varieties of different rational surfaces, such as S8 and D8?

Key findings

• The ideal of the secant variety σ₂(X) for the Segre embedding of Pⁿ₁×⋯×Pⁿₜ with t ≥ 3 is generated by the 3×3 minors of all flattenings of the associated tensor when n is sufficiently large.
• For the Segre-Veronese embedding of P¹×P¹×P¹, the ideal of σ₂(X) is generated by the 3×3 minors of a 4×4 matrix B, which is explicitly constructed from the homogeneous coordinates.
• The secant variety σ₂(S₉) of the Del Pezzo surface S₉ is defined by the 3×3 minors of a 4×4 matrix A₀, and this ideal is arithmetically Cohen-Macaulay with a known Eagon-Northcott resolution.
• The degree of σ₂(S₈) is 10, matching the degree of σ₂(D₈), and both varieties have identical graded Betti numbers, indicating indistinguishability by these numerical invariants.
• While σ₃(S₈) fills P⁸ as expected, σ₃(D₈) is a hypersurface defined by det(B) = 0, showing a key difference in higher secant behavior.
• The ideal of σ₃(S₉) is generated by the Aronhold invariant (a degree-4 form), confirming it is a hypersurface parameterizing Fermat cubics.