A characteristic free approach to secant varieties of triple Segre products.

*(English)*Zbl 1448.14052The aim of the paper is to study the ideal defining secant varieties of the Segre embedding of \(\mathbb{P}^1 \times \mathbb{P}^{a-1} \times \mathbb{P}^{b-1}\). The Cohen-Macaulay property of these secant varieties is also studied together with a bound on the Castelnuovo-Mumford regularity.

Given the Segre embedding \(\operatorname{Seg}(\mathbb{P}^1 \times \mathbb{P}^{a-1} \times \mathbb{P}^{b-1})\), it is of interest the study of its secant varieties \(\sigma_t (2,a,b)\) and their defining ideals denoted with \(I(a,b)^{\{t\}}\), or simply \(I(a,b)\) in the case \(t=1\). The unfolding of a tensor, sometimes known as flattening, is a transformation which reorganizes a tensor in a matrix. It is known that the minors of size \((t+1) \times (t+1)\) are part of the generators of \(I(a,b)^{\{t\}}\). The authors prove in this document that such minors suffices to generate this ideal in a characteristic free context. Tools borrowed from combinatorial commutative algebra such as secant ideals, simplicial complexes and the related Stanley-Reisner ideals are widely used.

At first they prove that the \(2 \times 2\) minors of the unfoldings on the second and on the third factor form a Gröbner basis of the ideal defining the Segre embedding \(\operatorname{Seg}(\mathbb{P}^1 \times \mathbb{P}^{a-1} \times \mathbb{P}^{b-1})\) with respect to any diagonal term order. After that, they show that the \((t+1) \times (t+1)\) minors of the unfoldings on the second and on the third factor are a Gröbner basis of the secant ideal \(I(a,b)^{\{t\}}\) defining \(\sigma_t(2,a,b)\) with respect to any diagonal term order. They also prove that the quotient \(R / I(a,b)^{\{t\}}\) is a Cohen–Macaulay domain. Such a result is achieved considering the simplicial complex associated to the initial ideal of \(I(a,b)^{\{t\}}\) and proving that it is shellable. A bound on the Castelnuovo-Mumford regularity of \(R / I(a,b)^{\{t\}}\) is also presented. They conclude conjecturing that \(R / I(2t,2t)^{\{t\}}\) is Gorenstein and they prove it for \(t \leq 3\).

Given the Segre embedding \(\operatorname{Seg}(\mathbb{P}^1 \times \mathbb{P}^{a-1} \times \mathbb{P}^{b-1})\), it is of interest the study of its secant varieties \(\sigma_t (2,a,b)\) and their defining ideals denoted with \(I(a,b)^{\{t\}}\), or simply \(I(a,b)\) in the case \(t=1\). The unfolding of a tensor, sometimes known as flattening, is a transformation which reorganizes a tensor in a matrix. It is known that the minors of size \((t+1) \times (t+1)\) are part of the generators of \(I(a,b)^{\{t\}}\). The authors prove in this document that such minors suffices to generate this ideal in a characteristic free context. Tools borrowed from combinatorial commutative algebra such as secant ideals, simplicial complexes and the related Stanley-Reisner ideals are widely used.

At first they prove that the \(2 \times 2\) minors of the unfoldings on the second and on the third factor form a Gröbner basis of the ideal defining the Segre embedding \(\operatorname{Seg}(\mathbb{P}^1 \times \mathbb{P}^{a-1} \times \mathbb{P}^{b-1})\) with respect to any diagonal term order. After that, they show that the \((t+1) \times (t+1)\) minors of the unfoldings on the second and on the third factor are a Gröbner basis of the secant ideal \(I(a,b)^{\{t\}}\) defining \(\sigma_t(2,a,b)\) with respect to any diagonal term order. They also prove that the quotient \(R / I(a,b)^{\{t\}}\) is a Cohen–Macaulay domain. Such a result is achieved considering the simplicial complex associated to the initial ideal of \(I(a,b)^{\{t\}}\) and proving that it is shellable. A bound on the Castelnuovo-Mumford regularity of \(R / I(a,b)^{\{t\}}\) is also presented. They conclude conjecturing that \(R / I(2t,2t)^{\{t\}}\) is Gorenstein and they prove it for \(t \leq 3\).

Reviewer: Reynaldo Staffolani (Trento)

##### MSC:

14N07 | Secant varieties, tensor rank, varieties of sums of powers |

13C40 | Linkage, complete intersections and determinantal ideals |

05E40 | Combinatorial aspects of commutative algebra |

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\textit{A. Conca} et al., Algebr. Comb. 3, No. 5, 1011--1021 (2020; Zbl 1448.14052)

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