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A characteristic free approach to secant varieties of triple Segre products. (English) Zbl 1448.14052
The 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 $$t \leq 3$$.
##### 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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