Abstract
Let I be either the ideal of maximal minors or the ideal of 2-minors of a row graded or column graded matrix of linear forms L. In previous work we showed that I is a Cartwright-Sturmfels ideal, that is, the multigraded generic initial ideal gin(I) of I is radical (and essentially independent of the term order chosen). In this paper we describe generators and prime decomposition of gin(I) in terms of data related to the linear dependences among the row or columns of the submatrices of L. In the case of 2-minors we also give a closed formula for its multigraded Hilbert series.
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References
C. Aholt, B. Sturmfels, R. Thomas, A Hilbert schemes in computer vision. Can. J. Math. 65(5), 961–988 (2013)
D. Bernstein, A. Zelevinsky, Combinatorics of maximal minors. J. Algebraic Combin. 2(2), 111–121 (1993)
A. Boocher, Free resolutions and sparse determinantal ideals. Math. Res. Lett. 19(4), 805–821 (2012)
D. Cartwright, B. Sturmfels, The Hilbert scheme of the diagonal in a product of projective spaces. Int. Math. Res. Not. 9, 1741–1771 (2010)
A. Conca, Linear spaces, transversal polymatroids and ASL domains. J. Algebraic Combin. 25(1), 25–41 (2007)
A. Conca, E. De Negri, E. Gorla, Universal Gröbner bases for maximal minors. Int. Math. Res. Not. 11, 3245–3262 (2015)
A. Conca, E. De Negri, E. Gorla, Universal Gröbner bases and Cartwright-Sturmfels ideals. Preprint (2016)
M.Y. Kalinin, Universal and comprehensive Gröbner bases of the classical determinantal ideal. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 373 (2009); Teoriya Predstavlenii, Dinamicheskie Sistemy, Kombinatornye Metody. XVII, 134–143, 348; translation in J. Math. Sci. (N.Y.) 168(3), 385–389 (2010)
B. Sturmfels, Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8 (American Mathematical Society, Providence, RI, 1996)
B. Sturmfels, A. Zelevinsky, Maximal minors and their leading terms. Adv. Math. 98(1), 65–112 (1993)
R. Villarreal, Monomial Algebras. Monographs and Textbooks in Pure and Applied Mathematics, vol. 238 (Dekker, New York, 2001)
Acknowledgements
The first and the second authors were partially supported by GNSAGA-INdAM. The third author was partially supported by the Swiss National Science Foundation under grant no. 200021_150207.
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Conca, A., Negri, E., Gorla, E. (2017). Multigraded Generic Initial Ideals of Determinantal Ideals. In: Conca, A., Gubeladze, J., Römer, T. (eds) Homological and Computational Methods in Commutative Algebra. Springer INdAM Series, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-61943-9_5
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DOI: https://doi.org/10.1007/978-3-319-61943-9_5
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