The pinched Veronese is Koszul
Giulio Caviglia
DOI: 10.1007/s10801-009-0176-1
Abstract
In this paper we prove that the coordinate ring of the pinched Veronese, k[ X 3, X 2 Y, XY 2, Y 3, X 2 Z, Y 2 Z, XZ 2, YZ 2, Z 3], is Koszul. The result is obtained by combining the use of a flat deformation induced by a distinguished weight together with a generalization of the notion of Koszul filtrations.
Pages: 539–548
Keywords: keywords Koszul algebras; pinched Veronese; Koszul filtration
Full Text: PDF
References
1. Anders, N.J.: CaTS, a software package for computing state polytopes of toric ideals, available from
2. Blum, S.: Subalgebras of bigraded Koszul algebras. J. Algebra 242(2), 795-809 (2001)
3. Bruns, W., Herzog, J., Vetter, U.: Syzygies and walks. In: Simis, A., Trung, N.V., Valla, G. (eds.) ICTP Proceedings `Commutative Algebra', pp. 36-57. World Scientific, Singapore (1994)
4. Bruns, W., Gubeladze, J., Trung, N.V.: Normal polytopes, triangulations, and Koszul algebras. J. Reine Angew. Math. 485, 123-160 (1997)
5. Conca, A., Herzog, J.: Castelnuovo-Mumford regularity of products of ideals. Preprint
6. Conca, A., Herzog, J., Trung, N.V., Valla, G.: Diagonal subalgebras of a bigraded algebras and embeddings of blow ups of projective spaces. Amer. J. Math. 119, 859-901 (1997)
7. Conca, A., Trung, N.V., Valla, G.: Koszul property for points in projective spaces. Math. Scand. 89, 201-216 (2001)
8. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, New York (1995)
9. Eisenbud, D., Reeves, A., Totaro, B.: Initial ideals, Veronese subrings, and rates of algebras. Adv. Math. 109, 168-187 (1994)
10. Fröberg, R.: Koszul algebras. In: Advances in Commutative Ring Theory (Fez, 1997). Lecture Notes in Pure and Appl. Math., vol. 205, pp. 337-350. Dekker, New York (1999)
11. Grayson, D., Stillman, M.: Macaulay 2: a software system for algebraic geometry and commutative algebra available over the web at
12. Herzog, J., Hibi, T., Restuccia, G.: Strongly Koszul algebras. Math. Scand. 86, 161-178 (2000)
13. Matsumura, H.: Commutative Ring Theory, 2nd edn. Cambridge Studies in Advanced Mathematics, vol.
8. Cambridge University Press, Cambridge (1989). Translated from the Japanese by M. Reid
14. Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol.
8. American Mathematical Society, Providence (1996), xii+162 pp.
2. Blum, S.: Subalgebras of bigraded Koszul algebras. J. Algebra 242(2), 795-809 (2001)
3. Bruns, W., Herzog, J., Vetter, U.: Syzygies and walks. In: Simis, A., Trung, N.V., Valla, G. (eds.) ICTP Proceedings `Commutative Algebra', pp. 36-57. World Scientific, Singapore (1994)
4. Bruns, W., Gubeladze, J., Trung, N.V.: Normal polytopes, triangulations, and Koszul algebras. J. Reine Angew. Math. 485, 123-160 (1997)
5. Conca, A., Herzog, J.: Castelnuovo-Mumford regularity of products of ideals. Preprint
6. Conca, A., Herzog, J., Trung, N.V., Valla, G.: Diagonal subalgebras of a bigraded algebras and embeddings of blow ups of projective spaces. Amer. J. Math. 119, 859-901 (1997)
7. Conca, A., Trung, N.V., Valla, G.: Koszul property for points in projective spaces. Math. Scand. 89, 201-216 (2001)
8. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, New York (1995)
9. Eisenbud, D., Reeves, A., Totaro, B.: Initial ideals, Veronese subrings, and rates of algebras. Adv. Math. 109, 168-187 (1994)
10. Fröberg, R.: Koszul algebras. In: Advances in Commutative Ring Theory (Fez, 1997). Lecture Notes in Pure and Appl. Math., vol. 205, pp. 337-350. Dekker, New York (1999)
11. Grayson, D., Stillman, M.: Macaulay 2: a software system for algebraic geometry and commutative algebra available over the web at
12. Herzog, J., Hibi, T., Restuccia, G.: Strongly Koszul algebras. Math. Scand. 86, 161-178 (2000)
13. Matsumura, H.: Commutative Ring Theory, 2nd edn. Cambridge Studies in Advanced Mathematics, vol.
8. Cambridge University Press, Cambridge (1989). Translated from the Japanese by M. Reid
14. Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol.
8. American Mathematical Society, Providence (1996), xii+162 pp.