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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.

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)
Full Text: DOI
[1] A. L. Gorodentsev, A. S. Khoroshkin, and A. N. Rudakov, ”On syzygies of highest weight orbits,” in: Amer. Math. Soc. Transl. (2), vol. 221, 2007, 79–120. · Zbl 1133.13012
[2] G. Ottaviani and R. Paoletti, ”Syzygies of Veronese embeddings,” Compositio Math., 125:1 (2001), 31–37; http://arxiv.org/abs/math/9811131 . · Zbl 0999.14016 · doi:10.1023/A:1002662809474
[3] A. Snowden, ”Syzygies of Segre embeddings and \(\Delta\)-modules,” Duke Math. J., 162:2 (2013), 225–277; http://arxiv.org/abs/1006.5248 . · Zbl 1279.13024 · doi:10.1215/00127094-1962767
[4] W. Fulton, Young Tableaux, Cambridge University Press, Cambridge, 1997.
[5] M. Green, ”Koszul cohomology and the geometry of projective varieties, I, II,” J. Differential Geom., 19,20 (1984), 125–171, 279–289. · Zbl 0559.14008
[6] S. I. Gel’fand and Yu. I. Manin, ”Homological algebra,” in: Algebra-5, Itogi Nauki i Tekhniki. Sovremennye Problemy Matematiki. Fundamental’nye Napravleniya, vol. 38, VINITI, Moscow, 1989, 5–233.
[7] G. Lancaster and J. Towber, ”Representation-functors and flag-algebras for the classical groups,” J. Algebra, 59:1 (1979), 16–38. · Zbl 0441.14013 · doi:10.1016/0021-8693(79)90149-2
[8] È. B. Vinberg and V. L. Popov, ”On a class of quasihomogeneous affine varieties,” Izv. Akad. Nauk SSSR Ser. Mat., 36:4 (1972), 749–764; English transl.: Math. USSR Izv., 6:4 (1972), 743–758.
[9] W. Fulton and J. Harris, Representation Theory. A First Course, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. · Zbl 0744.22001
[10] A. Polyshchuk and L. Positselski, Quadratic Algebras, Amer. Math. Soc., Providence, RI, 2005.
[11] J. Harris, Algebraic Geometry: A First Course, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1995. · Zbl 0779.14001
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