The reduction number and degree bound of projective subschemes

Cường, Đoàn; Kwak, Sijong

doi:10.1090/tran/7965

The reduction number and degree bound of projective subschemes
HTML articles powered by AMS MathViewer

by Đoàn Trung Cường and Sijong Kwak PDF

Trans. Amer. Math. Soc. 373 (2020), 1153-1180 Request permission

Abstract:

In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen–Macaulay with linear resolutions. Furthermore, it can be shown that there are only two types of reduced, irreducible projective varieties with almost maximal degree. We also give the possible explicit Betti tables for almost maximal cases. In addition, interesting examples are provided to understand our main results.

References

Jeaman Ahn and Sijong Kwak, Graded mapping cone theorem, multisecants and syzygies, J. Algebra 331 (2011), 243–262. MR 2774656, DOI 10.1016/j.jalgebra.2010.07.030
Jeaman Ahn and Sijong Kwak, On syzygies, degree, and geometric properties of projective schemes with property $\textbf {N}_{3,p}$, J. Pure Appl. Algebra 219 (2015), no. 7, 2724–2739. MR 3313504, DOI 10.1016/j.jpaa.2014.09.024
Alberto Alzati and Francesco Russo, On the $k$-normality of projected algebraic varieties, Bull. Braz. Math. Soc. (N.S.) 33 (2002), no. 1, 27–48. MR 1934282, DOI 10.1007/s005740200001
Alberto Alzati and José Carlos Sierra, A bound on the degree of schemes defined by quadratic equations, Forum Math. 24 (2012), no. 4, 733–750. MR 2949121, DOI 10.1515/form.2011.081
Isabel Bermejo and Philippe Gimenez, Computing the Castelnuovo-Mumford regularity of some subschemes of ${\Bbb P}_K^n$ using quotients of monomial ideals, J. Pure Appl. Algebra 164 (2001), no. 1-2, 23–33. Effective methods in algebraic geometry (Bath, 2000). MR 1854328, DOI 10.1016/S0022-4049(00)00143-2
M. Boij and J. Söderberg, Betti numbers of graded modules and the Multiplicity Conjecture in the non-Cohen–Macaulay case, Algebra Number Theory 6 (2012), no. 3, 437–454.
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
David Eisenbud and Shiro Goto, Linear free resolutions and minimal multiplicity, J. Algebra 88 (1984), no. 1, 89–133. MR 741934, DOI 10.1016/0021-8693(84)90092-9
David Eisenbud, Mark Green, Klaus Hulek, and Sorin Popescu, Restricting linear syzygies: algebra and geometry, Compos. Math. 141 (2005), no. 6, 1460–1478. MR 2188445, DOI 10.1112/S0010437X05001776
David Eisenbud and Frank-Olaf Schreyer, Betti numbers of graded modules and cohomology of vector bundles, J. Amer. Math. Soc. 22 (2009), no. 3, 859–888. MR 2505303, DOI 10.1090/S0894-0347-08-00620-6
Frank-Olaf Schreyer and David Eisenbud, Betti numbers of syzygies and cohomology of coherent sheaves, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 586–602. MR 2827810
Christopher A. Francisco, Huy Tài Hà, and Adam Van Tuyl, Splittings of monomial ideals, Proc. Amer. Math. Soc. 137 (2009), no. 10, 3271–3282. MR 2515396, DOI 10.1090/S0002-9939-09-09929-8
D. R. Grayson and M. E. Stillman, Macaulay $2$—A software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.
M. Green, “Generic initial ideals”, in J. Elias, J. M. Giral, R. M. Miró-Roig, and S. Zarzuela (eds.), Six lectures on commutative algebra, Progress in Mathematics, vol. 166, Birkhäuser, Basel, Switzerland, 1998, pp. 119–186.
Kangjin Han and Sijong Kwak, Analysis on some infinite modules, inner projection, and applications, Trans. Amer. Math. Soc. 364 (2012), no. 11, 5791–5812. MR 2946932, DOI 10.1090/S0002-9947-2012-05755-2
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Jürgen Herzog and Takayuki Hibi, Monomial ideals, Graduate Texts in Mathematics, vol. 260, Springer-Verlag London, Ltd., London, 2011. MR 2724673, DOI 10.1007/978-0-85729-106-6
Jürgen Herzog and Hema Srinivasan, Bounds for multiplicities, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2879–2902. MR 1458304, DOI 10.1090/S0002-9947-98-02096-0
Le Tuan Hoa, On minimal free resolutions of projective varieties of degree $=\textrm {codimension}+2$, J. Pure Appl. Algebra 87 (1993), no. 3, 241–250. MR 1228155, DOI 10.1016/0022-4049(93)90112-7
Hideyuki Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR 879273
Hirotsugu Naito, Minimal free resolution of curves of degree 6 or lower in the 3-dimensional projective space, Tokyo J. Math. 25 (2002), no. 1, 191–196. MR 1908222, DOI 10.3836/tjm/1244208945
Irena Peeva, Graded syzygies, Algebra and Applications, vol. 14, Springer-Verlag London, Ltd., London, 2011. MR 2560561, DOI 10.1007/978-0-85729-177-6
Ngô Việt Trung, Reduction exponent and degree bound for the defining equations of graded rings, Proc. Amer. Math. Soc. 101 (1987), no. 2, 229–236. MR 902533, DOI 10.1090/S0002-9939-1987-0902533-1
Wolmer V. Vasconcelos, Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series, vol. 195, Cambridge University Press, Cambridge, 1994. MR 1275840, DOI 10.1017/CBO9780511574726
Wolmer V. Vasconcelos, The reduction number of an algebra, Compositio Math. 104 (1996), no. 2, 189–197. MR 1421399