Counterexamples to the Eisenbud–Goto regularity conjecture

McCullough, Jason¹ ; Peeva, Irena²

¹ Mathematics Department, Iowa State University, Ames, Iowa 50011
² Mathematics Department, Cornell University, Ithaca, New York 14853

Journal of the American Mathematical Society, Tome 31 (2018) no. 2, pp. 473-496

Résumé

Our main theorem shows that the regularity of nondegenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field $k$. In particular, we provide counterexamples to the longstanding Regularity Conjecture, also known as the Eisenbud–Goto Conjecture (1984). We introduce a method which, starting from a homogeneous ideal $I$, produces a prime ideal whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of $I$. The method is also related to producing bounds in the spirit of Stillman’s Conjecture, recently solved by Ananyan and Hochster.

Détail

DOI : 10.1090/jams/891

Bibliographie

[1] Ananyan, Tigran, Hochster, Melvin Ideals generated by quadratic polynomials Math. Res. Lett. 2012 233 244

[2] Bayer, Dave, Mumford, David What can be computed in algebraic geometry? 1993 1 48

[3] Bayer, David, Stillman, Michael On the complexity of computing syzygies J. Symbolic Comput. 1988 135 147

[4] Becker, Joseph On the boundedness and the unboundedness of the number of generators of ideals and multiplicity J. Algebra 1977 447 453

[5] Beder, Jesse, Mccullough, Jason, Nãºã±Ez-Betancourt, Luis, Seceleanu, Alexandra, Snapp, Bart, Stone, Branden Ideals with larger projective dimension and regularity J. Symbolic Comput. 2011 1105 1113

[6] Bertram, Aaron, Ein, Lawrence, Lazarsfeld, Robert Vanishing theorems, a theorem of Severi, and the equations defining projective varieties J. Amer. Math. Soc. 1991 587 602

[7] Brodmann, Markus Cohomology of certain projective surfaces with low sectional genus and degree 1999 173 200

[8] Brodmann, M., Vogel, W. Bounds for the cohomology and the Castelnuovo regularity of certain surfaces Nagoya Math. J. 1993 109 126

[9] Caviglia, Giulio, Sbarra, Enrico Characteristic-free bounds for the Castelnuovo-Mumford regularity Compos. Math. 2005 1365 1373

[10] Chardin, Marc Some results and questions on Castelnuovo-Mumford regularity 2007 1 40

[11] Chardin, Marc Bounds for Castelnuovo-Mumford regularity in terms of degrees of defining equations 2003 67 73

[12] Chardin, Marc, Fall, Amadou Lamine Sur la régularité de Castelnuovo-Mumford des idéaux, en dimension 2 C. R. Math. Acad. Sci. Paris 2005 233 238

[13] Chardin, Marc, Ulrich, Bernd Liaison and Castelnuovo-Mumford regularity Amer. J. Math. 2002 1103 1124

[14] De Fernex, Tommaso, Ein, Lawrence A vanishing theorem for log canonical pairs Amer. J. Math. 2010 1205 1221

[15] Derksen, Harm, Sidman, Jessica A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements Adv. Math. 2002 151 157

[16] Eisenbud, David Lectures on the geometry of syzygies 2004 115 152

[17] Eisenbud, David The geometry of syzygies 2005

[18] Eisenbud, David Commutative algebra 1995

[19] Eisenbud, David, Goto, Shiro Linear free resolutions and minimal multiplicity J. Algebra 1984 89 133

[20] Eisenbud, David, Ulrich, Bernd Notes on regularity stabilization Proc. Amer. Math. Soc. 2012 1221 1232

[21] Flenner, Hubert Die Sätze von Bertini für lokale Ringe Math. Ann. 1977 97 111

[22] Flã¸Ystad, Gunnar, Mccullough, Jason, Peeva, Irena Three themes of syzygies Bull. Amer. Math. Soc. (N.S.) 2016 415 435

[23] Galligo, Andr㩠Théorème de division et stabilité en géométrie analytique locale Ann. Inst. Fourier (Grenoble) 1979

[24] Gasharov, Vesselin, Peeva, Irena, Welker, Volkmar The lcm-lattice in monomial resolutions Math. Res. Lett. 1999 521 532

[25] Giaimo, Daniel On the Castelnuovo-Mumford regularity of connected curves Trans. Amer. Math. Soc. 2006 267 284

[26] Giusti, M. Some effectivity problems in polynomial ideal theory 1984 159 171

[27] Gruson, L., Lazarsfeld, R., Peskine, C. On a theorem of Castelnuovo, and the equations defining space curves Invent. Math. 1983 491 506

[28] Herzog, Jã¼Rgen, Hibi, Takayuki Castelnuovo-Mumford regularity of simplicial semigroup rings with isolated singularity Proc. Amer. Math. Soc. 2003 2641 2647

[29] Hoa, L㪠Tuã¢N, Miyazaki, Chikashi Bounds on Castelnuovo-Mumford regularity for generalized Cohen-Macaulay graded rings Math. Ann. 1995 587 598

[30] Huneke, Craig On the symmetric and Rees algebra of an ideal generated by a 𝑑-sequence J. Algebra 1980 268 275

[31] Koh, Jee Ideals generated by quadrics exhibiting double exponential degrees J. Algebra 1998 225 245

[32] Kustin, Andrew R., Polini, Claudia, Ulrich, Bernd Rational normal scrolls and the defining equations of Rees algebras J. Reine Angew. Math. 2011 23 65

[33] Kwak, Sijong Castelnuovo regularity for smooth subvarieties of dimensions 3 and 4 J. Algebraic Geom. 1998 195 206

[34] Kwak, Si-Jong Castelnuovo-Mumford regularity bound for smooth threefolds in 𝑃⁵ and extremal examples J. Reine Angew. Math. 1999 21 34

[35] Lazarsfeld, Robert A sharp Castelnuovo bound for smooth surfaces Duke Math. J. 1987 423 429

[36] Lazarsfeld, Robert Positivity in algebraic geometry. I 2004

[37] Mayr, Ernst W., Meyer, Albert R. The complexity of the word problems for commutative semigroups and polynomial ideals Adv. in Math. 1982 305 329

[38] Mccullough, Jason, Seceleanu, Alexandra Bounding projective dimension 2013 551 576

[39] Nagata, Masayoshi Local rings 1962

[40] Niu, Wenbo Castelnuovo-Mumford regularity bounds for singular surfaces Math. Z. 2015 609 620

[41] Noma, Atsushi Generic inner projections of projective varieties and an application to the positivity of double point divisors Trans. Amer. Math. Soc. 2014 4603 4623

[42] Peeva, Irena Graded syzygies 2011

[43] Pinkham, Henry C. A Castelnuovo bound for smooth surfaces Invent. Math. 1986 321 332

[44] Ran, Ziv Local differential geometry and generic projections of threefolds J. Differential Geom. 1990 131 137

[45] Ullery, Brooke Designer ideals with high Castelnuovo-Mumford regularity Math. Res. Lett. 2014 1215 1225