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Syzygy algebras for Segre embeddings. (English. Russian original) Zbl 1326.13011
Funct. Anal. Appl. 47, No. 3, 210-226 (2013); translation from Funkts. Anal. Prilozh. 47, No. 3, 54-74 (2013).
Let \(U\) and \(V\) be vector spaces over a field \(k\), \(\dim U = m\), \(\dim V = n\). The field \(k\) is supposed to have zero characteristic and to be algebraically closed. Consider the tensor product \(U\otimes V\). Let \(M\subset U \otimes V\) be the subset of decomposable tensors. Note that \(M \) is a cone. It is well known that \(\mathbb{P}(M) = \mathbb{P}(U)\times \mathbb{P}(V )\subset \mathbb{P}(U \otimes V )\). This embedding is called the Segre embedding. One can identify \(U \otimes V\) with the space of \(m \times n\)-matrices. Then \(M\) gets identified with the set of matrices of rank \(\leq 1\). Its ideal \(I(M)\) is generated by all \(2 \times 2\)-minors as polynomials in matrix elements. There are relations between these generators. If we take minimal sets of generators and relations, then they will generate so called syzygy spaces. In general, it is very difficult to find syzygy spaces. They are not completely described even for projective curves.
In this paper, the author considers the Segre embedding of the product of two projective spaces. The representation of the group \(G = \mathrm{GL}(U) \times \mathrm{GL}(V )\) on syzygy spaces of this embedding are described in Theorem 1.2.
The main goal of this paper is to describe the syzygy spaces of the Segre embedding and syzygy spaces of sheaves \(\mathcal{O}_{\mathbb{P}(U)\times \mathbb{P}(V )}(a, b)\) on \(\mathbb{P}(U \otimes V )\) for \(a \geq -\dim U\) and \(b \geq -\dim V\) (see Theorem 4.9 and Definition 2.11) as \(G\)-modules.
In section 4, the author uses combinatorics from Section 3 to calculate syzygies of the Segre embedding and to calculate minimal resolutions of sheaves. The appendices are devoted to examples.

MSC:
13D02 Syzygies, resolutions, complexes and commutative rings
14E25 Embeddings in algebraic geometry
13D09 Derived categories and commutative rings
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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