Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Syzygies and singularities of tensor product surfaces of bidegree $ (2,1)$


Authors: Hal Schenck, Alexandra Seceleanu and Javid Validashti
Journal: Math. Comp. 83 (2014), 1337-1372
MSC (2010): Primary 14M25; Secondary 14F17
DOI: https://doi.org/10.1090/S0025-5718-2013-02764-0
Published electronically: August 14, 2013
MathSciNet review: 3167461
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ U \subseteq H^0({\mathcal {O}_{\mathbb{P}^1 \times \mathbb{P}^1}}(2,1))$ be a basepoint free four-dimensional vector space. The sections corresponding to $ U$ determine a regular map $ \phi _U: {\mathbb{P}^1 \times \mathbb{P}^1} \longrightarrow \mathbb{P}^3$. We study the associated bigraded ideal $ I_U \subseteq k[s,t;u,v]$ from the standpoint of commutative algebra, proving that there are exactly six numerical types of possible bigraded minimal free resolution. These resolutions play a key role in determining the implicit equation for $ \phi _U({\mathbb{P}^1 \times \mathbb{P}^1})$, via work of Busé-Jouanolou, Busé-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex $ \mathcal {Z}$. In four of the six cases $ I_U$ has a linear first syzygy; remarkably from this we obtain all differentials in the minimal free resolution. In particular, this allows us to explicitly describe the implicit equation and singular locus of the image.


References [Enhancements On Off] (What's this?)

  • [1] Annetta Aramova, Kristina Crona, and Emanuela De Negri, Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions, J. Pure Appl. Algebra 150 (2000), no. 3, 215-235. MR 1769356 (2001f:13046), https://doi.org/10.1016/S0022-4049(99)00100-0
  • [2] Nicolás Botbol, The implicit equation of a multigraded hypersurface, J. Algebra 348 (2011), 381-401. MR 2852248 (2012m:14094), https://doi.org/10.1016/j.jalgebra.2011.09.019
  • [3] Nicolás Botbol, Alicia Dickenstein, and Marc Dohm, Matrix representations for toric parametrizations, Comput. Aided Geom. Design 26 (2009), no. 7, 757-771. MR 2569833 (2010j:65023), https://doi.org/10.1016/j.cagd.2009.03.005
  • [4] David A. Buchsbaum and David Eisenbud, What makes a complex exact?, J. Algebra 25 (1973), 259-268. MR 0314819 (47 #3369)
  • [5] Laurent Busé and Jean-Pierre Jouanolou, On the closed image of a rational map and the implicitization problem, J. Algebra 265 (2003), no. 1, 312-357. MR 1984914 (2004e:14024), https://doi.org/10.1016/S0021-8693(03)00181-9
  • [6] Laurent Busé and Marc Chardin, Implicitizing rational hypersurfaces using approximation complexes, J. Symbolic Comput. 40 (2005), no. 4-5, 1150-1168. MR 2172855 (2006g:14097), https://doi.org/10.1016/j.jsc.2004.04.005
  • [7] Marc Chardin, Implicitization using approximation complexes, Algebraic geometry and geometric modeling, Math. Vis., Springer, Berlin, 2006, pp. 23-35. MR 2279841 (2007j:14097), https://doi.org/10.1007/978-3-540-33275-6_2
  • [8] Ciro Ciliberto, Francesco Russo, and Aron Simis, Homaloidal hypersurfaces and hypersurfaces with vanishing Hessian, Adv. Math. 218 (2008), no. 6, 1759-1805. MR 2431661 (2009j:14056), https://doi.org/10.1016/j.aim.2008.03.025
  • [9] David A. Cox, The moving curve ideal and the Rees algebra, Theoret. Comput. Sci. 392 (2008), no. 1-3, 23-36. MR 2394983 (2009a:13003), https://doi.org/10.1016/j.tcs.2007.10.012
  • [10] David Cox, Curves, surfaces, and syzygies, Topics in algebraic geometry and geometric modeling, Contemp. Math., vol. 334, Amer. Math. Soc., Providence, RI, 2003, pp. 131-150. MR 2039970 (2005g:14113), https://doi.org/10.1090/conm/334/05979
  • [11] David Cox, Alicia Dickenstein, and Hal Schenck, A case study in bigraded commutative algebra, Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math., vol. 254, Chapman & Hall/CRC, Boca Raton, FL, 2007, pp. 67-111. MR 2309927 (2008d:13018), https://doi.org/10.1201/9781420050912.ch3
  • [12] David Cox, Ronald Goldman, and Ming Zhang, On the validity of implicitization by moving quadrics of rational surfaces with no base points, J. Symbolic Comput. 29 (2000), no. 3, 419-440. MR 1751389 (2002d:14098), https://doi.org/10.1006/jsco.1999.0325
  • [13] W. L. F. Degen, The types of rational $ (2,1)$-Bézier surfaces, Comput. Aided Geom. Design 16 (1999), no. 7, 639-648. Dedicated to Paul de Faget de Casteljau. MR 1718055 (2000h:65030), https://doi.org/10.1016/S0167-8396(99)00028-X
  • [14] Alicia Dickenstein and Ioannis Z. Emiris, Multihomogeneous resultant formulae by means of complexes, J. Symbolic Comput. 36 (2003), no. 3-4, 317-342. International Symposium on Symbolic and Algebraic Computation (ISSAC'2002) (Lille). MR 2004032 (2004h:13029), https://doi.org/10.1016/S0747-7171(03)00086-5
  • [15] Marc Dohm, Implicitization of rational ruled surfaces with $ \mu $-bases, J. Symbolic Computation 44 (2009), 479-489. MR 2499923 (2010b:14120)
  • [16] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960 (97a:13001)
  • [17] David Eisenbud, Craig Huneke, and Wolmer Vasconcelos, Direct methods for primary decomposition, Invent. Math. 110 (1992), no. 2, 207-235. MR 1185582 (93j:13032), https://doi.org/10.1007/BF01231331
  • [18] W. Edge, The theory of ruled surfaces, Cambridge University Press, 1931.
  • [19] M. Elkadi, A. Galligo and T. H. Lê, Parametrized surfaces in $ \mathbb{P}^3$ of bidegree $ (1,2)$, Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2004, 141-148.
  • [20] Thi-Ha Lê and André Galligo, General classification of $ (1,2)$ parametric surfaces in $ \mathbb{P}^3$, Geometric modeling and algebraic geometry, Springer, Berlin, 2008, pp. 93-113. MR 2381606 (2009a:14075), https://doi.org/10.1007/978-3-540-72185-7_6
  • [21] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, Mathematics: Theory & Applications, Birkhäuser Boston Inc., Boston, MA, 1994. MR 1264417 (95e:14045)
  • [22] Joe Harris, Algebraic geometry, Graduate Texts in Mathematics, vol. 133, Springer-Verlag, New York, 1992. A first course. MR 1182558 (93j:14001)
  • [23] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157 (57 #3116)
  • [24] J. Herzog, A. Simis, and W. V. Vasconcelos, Approximation complexes of blowing-up rings, J. Algebra 74 (1982), no. 2, 466-493. MR 647249 (83h:13023), https://doi.org/10.1016/0021-8693(82)90034-5
  • [25] J. Herzog, A. Simis, and W. V. Vasconcelos, Approximation complexes of blowing-up rings. II, J. Algebra 82 (1983), no. 1, 53-83. MR 701036 (85b:13015), https://doi.org/10.1016/0021-8693(83)90173-4
  • [26] Juan C. Migliore, Introduction to liaison theory and deficiency modules, Progress in Mathematics, vol. 165, Birkhäuser Boston Inc., Boston, MA, 1998. MR 1712469 (2000g:14058)
  • [27] G. Salmon, Traité de Géométrie analytique a trois dimensiones, Paris, Gauthier-Villars, 1882.
  • [28] T. W. Sederberg, F. Chen, Implicitization using moving curves and surfaces, in Proceedings of SIGGRAPH, 1995, 301-308.
  • [29] Tom Sederberg, Ron Goldman, and Hang Du, Implicitizing rational curves by the method of moving algebraic curves, J. Symbolic Comput. 23 (1997), no. 2-3, 153-175. Parametric algebraic curves and applications (Albuquerque, NM, 1995). MR 1448692 (98g:14072), https://doi.org/10.1006/jsco.1996.0081
  • [30] Thomas W. Sederberg, Takafumi Saito, Dong Xu Qi, and Krzysztof S. Klimaszewski, Curve implicitization using moving lines, Comput. Aided Geom. Design 11 (1994), no. 6, 687-706. MR 1305914 (95h:65011), https://doi.org/10.1016/0167-8396(94)90059-0
  • [31] S. Zubė, Correspondence and $ (2,1)$-Bézier surfaces, Liet. Mat. Rink. 43 (2003), no. 1, 99-122 (English, with English and Lithuanian summaries); English transl., Lithuanian Math. J. 43 (2003), no. 1, 83-102. MR 1996756 (2004k:14061), https://doi.org/10.1023/A:1022971323662
  • [32] S. Zubė, Bidegree $ (2,1)$ parametrizable surfaces in projective $ 3$-space, Liet. Mat. Rink. 38 (1998), no. 3, 379-402 (English, with English and Lithuanian summaries); English transl., Lithuanian Math. J. 38 (1998), no. 3, 291-308 (1999). MR 1657885 (99m:51030), https://doi.org/10.1007/BF02465903

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 14M25, 14F17

Retrieve articles in all journals with MSC (2010): 14M25, 14F17


Additional Information

Hal Schenck
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: schenck@math.uiuc.edu

Alexandra Seceleanu
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
Email: aseceleanu2@math.unl.edu

Javid Validashti
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: jvalidas@illinois.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02764-0
Keywords: Tensor product surface, bihomogeneous ideal, Segre-Veronese map
Received by editor(s): February 25, 2012
Received by editor(s) in revised form: October 30, 2012, and November 1, 2012
Published electronically: August 14, 2013
Additional Notes: The first author was supported by NSF 1068754, NSA H98230-11-1-0170
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society