×

Four counterexamples in combinatorial algebraic geometry. (English) Zbl 1018.14022

From the paper: We present counterexamples to four conjectures which appeared in the literature in commutative algebra and algebraic geometry. The four questions to be studied are largely unrelated, and yet our answers are connected by a common thread: They are combinatorial in nature, involving monomial ideals and binomial ideals, and they were found by exhaustive computer search using the symbolic algebra systems Maple and Macaulay 2.
In Section 1 we answer K. Chandler’s question [Commun. Algebra 12, 3773-3776 (1997; Zbl 0928.14033)]whether the Castelnuovo-Mumford regularity of a homogeneous polynomial ideal \(I\) satisfies the inequality \(\text{reg}(I^r)\leq r \cdot \text{reg}(I)\). In section 2 we settle a conjecture published two decades ago by J. Briançon and A. Iarrobino [J. Algebra 55, 536-544 (1978; Zbl 0402.14003)], by showing that the most singular point on the Hilbert scheme of points need not be the monomial ideal with most generators. In section 3 we construct a smooth projectively normal curve which is defided by quadrics but is not Koszul; this solves a problem posed by D. C. Butler [J. Differ. Geom. 39, 1-34 (1994; Zbl 0808.14024) problem 6.5]and A. Polishchuk [J. Algebra 178, 122-135 (1995; Zbl 0861.14030), p. 123]. Section 4 disproves an overly optimistic conjecture of mine [B. Sturmfels, “Gröbner bases and convex polytopes” (1995; Zbl 0856.13020) example 13.17]about the Gröbner bases of a certain toric 4-fold.

MSC:

14Q99 Computational aspects in algebraic geometry
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

Software:

Macaulay2; Maple
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Berman, D., The number of generators of a colength \(N\) ideal in a power series ring, J. Algebra, 73, 156-166 (1981) · Zbl 0491.13013
[2] Briançon, J.; Iarrobino, A., Dimension of the punctual Hilbert scheme, J. Algebra, 55, 536-544 (1978) · Zbl 0402.14003
[3] Butler, D. C., Normal generation of vector bundles over a curve, J. Differential Geom., 39, 1-34 (1994) · Zbl 0808.14024
[4] Chandler, K., Regularity of the powers of an ideal, Comm. Algebra, 12, 3773-3776 (1997) · Zbl 0928.14033
[6] Cox, D., The homogeneous coordinate ring of a toric variety, J. Algebraic Geom., 4, 17-50 (1995) · Zbl 0846.14032
[7] Cutkosky, D.; Herzog, J.; Trung, N. V., Asymptotic behavior of the Castelnuovo-Mumford regularity, Compositio Math., 118, 243-261 (1999) · Zbl 0974.13015
[8] Eagon, J. A.; Reiner, V., Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra, 130, 265-275 (1998) · Zbl 0941.13016
[9] Eisenbud, D.; Reeves, A.; Totaro, B., Initial ideals, Veronese subrings, and rates of algebras, Adv. Math., 109, 168-187 (1994) · Zbl 0839.13013
[10] Geramita, A. V.; Gimigliano, A.; Pitteloud, Y., Graded Betti numbers of some embedded rational \(n\)-folds, Math. Ann., 301, 363-389 (1995) · Zbl 0813.14008
[12] Hoa, L. T.; Trung, N. V., On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals, Math. Z., 229, 519-537 (1998) · Zbl 0931.13015
[13] Iarrobino, A., Reducibility of the families of 0-dimensional schemes on a variety, Invent. Math., 15, 72-77 (1972) · Zbl 0227.14006
[14] Kodiyalam, V., Asymptotic behaviour of Castelnuovo-Mumford regularity, Proc. Amer. Math. Soc., 128, 407-411 (2000) · Zbl 0929.13004
[15] Ohsugi, H.; Hibi, T., Toric ideals generated by quadratic binomials, J. Algebra, 215, 509-527 (1999) · Zbl 0943.13014
[16] Polishchuk, A., On the Koszul property of the homogeneous coordinate ring of a curve, J. Algebra, 178, 122-135 (1995) · Zbl 0861.14030
[17] Reeves, A.; Stillman, M., Smoothness of the lexicographic point, J. Algebraic Geom., 6, 235-246 (1997) · Zbl 0924.14004
[18] Roos, J.-E.; Sturmfels, B., A toric ring with irrational Poincaré-Betti series, C. R. Acad. Sci. Paris Sér. I Math., 326, 141-146 (1998) · Zbl 0934.14033
[20] Smith, K.; Swanson, I., Linear bounds on growth of associated primes, Comm. Algebra, 25, 3071-3079 (1997) · Zbl 0889.13002
[21] Stanley, R., Theory and applications of plane partitions, I, II, Stud. Appl. Math., 50, 167-188 (1971) · Zbl 0225.05011
[22] Stanley, R., Combinatorics and Commutative Algebra. Combinatorics and Commutative Algebra, Progress in Mathematics, 41 (1996), Birkhäuser: Birkhäuser Boston · Zbl 0838.13008
[23] Sturmfels, B., Gröbner Bases and Convex Polytopes. Gröbner Bases and Convex Polytopes, University Lecture Notes, 8 (1995), Am. Math. Soc: Am. Math. Soc Providence
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.