Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals

References

• [Bayer and Stillman 87] D. Bayer and M. Stillman. “A Theorem on Refining Division Orders by the Reverse Lexicographic Order.” Duke Math. J., 55:2 (1987), 321–328.
   Web of Science ®Google Scholar
• [Bertone 15] C. Bertone. “Quasi-stable Ideals and Borel-fixed Ideals With a Given Hilbert Polynomial.” Appl. Algebra Engrg. Comm. Comput., 26:6, (2015), 507–525 .
   Web of Science ®Google Scholar
• [Bertone et al. 12] C. Bertone, F. Cioffi, and M. Roggero. “A Division Algorithm in an Affine Framework for Flat Families Covering Hilbert Schemes.” Available at arXiv:1211.7264v1, 2012.
   Google Scholar
• [Bertone et al. 17a] C. Bertone, F. Cioffi, and M. Roggero. “Macaulay-like Marked Bases.” J. Algebra Appl., 16:5 (2017a), 1750100, 36.
   Web of Science ®Google Scholar
• [Bertone et al. 17b] C. Bertone, F. Cioffi, and M. Roggero. “Double-generic Initial Ideal and Hilbert Scheme.” Ann. Mat. Pura Appl. (4), 196:1 (2017b), 19–41.
   Web of Science ®Google Scholar
• [Bruns and Herzog 93] W. Bruns and J. Herzog. Cohen-Macaulay rings, volume 39 of Cambridge Studies in Advanced Mathematics. Cambridge: Cambridge University Press, 1993. ISBN 0-521-41068-1.
   Google Scholar
• [Buczyński and Jelisiejew 17] J. Buczyński and J. Jelisiejew. “Finite Schemes and Secant Varieties Over Arbitrary Characteristic.” Differ. Geom. Appl., 55(2017), 13–67, .
   Google Scholar
• [Cartwright et al. 09] D. A. Cartwright, D. Erman, M. Velasco, and B. Viray. “Hilbert Schemes of 8 Points.” Algebra Number Theory, 3:7 (2009), 763–795.
   Web of Science ®Google Scholar
• [Casnati and Notari 09] G. Casnati and R. Notari. “On the Gorenstein Locus of Some Punctual Hilbert Schemes.” J. Pure Appl. Algebra, 213:11 (2009):2055–2074.
   Web of Science ®Google Scholar
• [Casnati and Notari 11] G. Casnati and R. Notari. “On the Irreducibility and the Singularities of the Gorenstein Locus of the Punctual Hilbert Scheme of Degree 10.” J. Pure Appl. Algebra, 215:6 (2011), 1243–1254.
   Web of Science ®Google Scholar
• [Casnati and Notari 14] G. Casnati and R. Notari. “On the Gorenstein Locus of the Punctual Hilbert Scheme of Degree 11.” J. Pure Appl. Algebra, 218:9 (2014), 1635–1651, .
   Web of Science ®Google Scholar
• [Casnati et al. 11] G. Casnati, J. Jelisiejew, and R. Notari. “Irreducibility of the Gorenstein Loci of Hilbert Schemes Via Ray Families.” Algebra Number Theory, 9:7 (2015), 1525–1570.
   Web of Science ®Google Scholar
• [Ceria et al. 15] M. Ceria, T. Mora, and M. Roggero. “Term-ordering Free Involutive Bases.” J. Symbolic Comput., 68:part 2 (2015), 87–108.
   Google Scholar
• [Cioffi et al. 11] F. Cioffi, P. Lella, M. G. Marinari, and M. Roggero. “Segments and Hilbert Schemes of Points.” Discret. Math., 311 (2011), 2238–2252.
   Web of Science ®Google Scholar
• [Eisenbud 95] D. Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics. New York: Springer-Verlag, 1995. With a view toward algebraic geometry.
   Google Scholar
• [Elias and Rossi 12] J. Elias and M. E. Rossi. “Isomorphism Classes of Short Gorenstein Local Rings via Macaulay’s Inverse System.” Trans. Amer. Math. Soc., 364:9 (2012), 4589–4604.
   Web of Science ®Google Scholar
• [Ferrarese and Roggero 09] G. Ferrarese and M. Roggero. “Homogeneous Varieties for Hilbert Schemes.” Int. J. Algebra, 3:9–12 (2009), 547–557.
   Google Scholar
• [Fogarty 68] J. Fogarty. “Algebraic Families on an Algebraic Surface.” Amer. J. Math, 90(1968), 511–521.
   Google Scholar
• [Greco and Marinari 78] S. Greco and M. G. Marinari. “Nagata’s Criterion and Openness of Loci for Gorenstein and Complete Intersection.” Math. Z., 160:3 (1978), 207–216.
   Web of Science ®Google Scholar
• [Huibregtse 07] M. E. Huibregtse. “Some Elementary Components of the Hilbert Scheme of Points.” Rocky Mountain J. Math., 47:4 ()1169–1225.
   Web of Science ®Google Scholar
• [Iarrobino and Emsalem 78] A. Iarrobino and J. Emsalem. “Some Zero-dimensional Generic Singularities; Finite Algebras Having Small Tangent Space.” Compositio Math., 36:2 (1978), 145–188.
   Web of Science ®Google Scholar
• [Iarrobino and Kanew 99] A. Iarrobino and V. Kanev. Power Sums, Gorenstein Algebras, and Determinantal Loci, volume 1721 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1999. Appendix C by Iarrobino and Steven L. Kleiman.
   Google Scholar
• [Jelisiejew 14] J. Jelisiejew. “Local Finite-dimensional Gorenstein k-algebras Having Hilbert Function (1,5,5,1) are Smoothable.” J. Algebra Appl., 13:8 (2014), 1450056, 7.
   Web of Science ®Google Scholar
• [Jelisiejew 17] J. Jelisiejew. “Classifying Local Artinian Gorenstein Algebras.” Collect. Math., 68:1 (2017), 101–127.
   Web of Science ®Google Scholar
• [Jelisiejew 19] J. Jelisiejew. “Elementary Components of Hilbert Schemes.” Journal of the London Mathematical Society, 2019, doi: https://doi.org/10.1112/jlms.12212
   Google Scholar
• [Kreuzer and Robbiano 00] M. Kreuzer and L. Robbiano. Computational Commutative Algebra. 1. Berlin: Springer-Verlag, 2000.
   Google Scholar
• [Kreuzer and Robbiano 05] M. Kreuzer and L. Robbiano. Computational Commutative Algebra. 2. Berlin: Springer-Verlag, 2005.
   Google Scholar
• [Lella 12] P. Lella. “An Efficient Implementation of the Algorithm Computing the Borel-fixed Points of a Hilbert Scheme.” ISSAC 2012—Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, pages 242–248. ACM, New York, 2012.
   Google Scholar
• [Lella and Roggero 11] P. Lella and M. Roggero. “Rational Components of Hilbert Schemes.” Rendiconti del Seminario Matematico dell’Università di Padova, 126(2011), 11–45.
   Google Scholar
• [Miller and Sturmfels 05] E. Miller and B. Sturmfels. Combinatorial Commutative Algebra, volume 227 of Graduate Texts in Mathematics. New York: Springer-Verlag, 2005.
   Google Scholar
• [Mora 05] T. Mora. Solving Polynomial Equation Systems. II, volume 99 of Encyclopedia of Mathematics and its Applications. Cambridge: Cambridge University Press, 2005. Macaulay’s paradigm and Gröbner technology.
   Google Scholar
• [Reeves and Sturmfels 93] A. Reeves and B. Sturmfels. “A Note on Polynomial Reduction.” J. Symbolic Comput., 16:3 (1993), 273–277.
   Web of Science ®Google Scholar
• [Seiler 09] W. M. Seiler. “A Combinatorial Approach to Involution and δ-regularity. I. Involutive Bases in Polynomial Algebras of Solvable Type.” Appl. Algebra Engrg. Comm. Comput., 20:3/4 (2009), 207–259.
   Web of Science ®Google Scholar
• [Seiler 09] Werner M. Seiler. “A Combinatorial Approach to Involution and δ-regularity. II. Structure Analysis of Polynomial Modules with Pommaret Bases.” Appl. Algebra Engrg. Comm. Comput., 20:3/4 (2009b), 261–338.
   Web of Science ®Google Scholar
• [Stoia 75] M. Stoia. “Points de Gorenstein d’un morphisme.” C. R. Acad. Sci. Paris Sér. A-B, 281:20 (1975), Aii, A847–A849.
   Google Scholar