Tensor product surfaces and linear syzygies. (English) Zbl 1327.14225
Summary: Let \(U \subseteq H^0(\mathcal{O}_{\mathbb P^1 \times \mathbb P^1}(a,b))\) be a basepoint free four-dimensional vector space, with \(a,b \geq 2\). The sections corresponding to \(U\) determine a regular map \(\phi_U: {\mathbb P^1 \times \mathbb P^1}\longrightarrow \mathbb P^3\). We show that there can be at most one linear syzygy on the associated bigraded ideal \(I_U \subseteq k[s,t;u,v]\). Existence of a linear syzygy, coupled with the assumption that \(U\) is basepoint free, implies the existence of an additional “special pair” of minimal first syzygies. Using results of Botbol, we show that these three syzygies are sufficient to determine the implicit equation of \(\phi _U(\mathbb P^1 \times \mathbb P^1)\), and that \(\mathrm{Sing}(\phi _U(\mathbb P^1 \times \mathbb P^1))\) contains a line.
MSC:
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 14N05 | Projective techniques in algebraic geometry |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 14F17 | Vanishing theorems in algebraic geometry |
References:
| [1] | Botbol, Nicol{\'a}s, The implicit equation of a multigraded hypersurface, J. Algebra, 348, 381-401 (2011) · Zbl 1242.14049 · doi:10.1016/j.jalgebra.2011.09.019 |
| [2] | Botbol, Nicol{\'a}s; Dickenstein, Alicia; Dohm, Marc, Matrix representations for toric parametrizations, Comput. Aided Geom. Design, 26, 7, 757-771 (2009) · Zbl 1205.65070 · doi:10.1016/j.cagd.2009.03.005 |
| [3] | Bus{\'e}, Laurent; Jouanolou, Jean-Pierre, On the closed image of a rational map and the implicitization problem, J. Algebra, 265, 1, 312-357 (2003) · Zbl 1050.13010 · doi:10.1016/S0021-8693(03)00181-9 |
| [4] | Bus{\'e}, Laurent; Chardin, Marc, Implicitizing rational hypersurfaces using approximation complexes, J. Symbolic Comput., 40, 4-5, 1150-1168 (2005) · Zbl 1120.14052 · doi:10.1016/j.jsc.2004.04.005 |
| [5] | Chardin, Marc, Implicitization using approximation complexes. Algebraic geometry and geometric modeling, Math. Vis., 23-35 (2006), Springer, Berlin · Zbl 1116.14049 · doi:10.1007/978-3-540-33275-6\_2 |
| [6] | Cox, David A., The moving curve ideal and the Rees algebra, Theoret. Comput. Sci., 392, 1-3, 23-36 (2008) · Zbl 1170.13004 · doi:10.1016/j.tcs.2007.10.012 |
| [7] | Cox, David, Curves, surfaces, and syzygies. Topics in algebraic geometry and geometric modeling, Contemp. Math. 334, 131-150 (2003), Amer. Math. Soc., Providence, RI · Zbl 1058.14072 · doi:10.1090/conm/334/05979 |
| [8] | Cox, David; Dickenstein, Alicia; Schenck, Hal, A case study in bigraded commutative algebra. Syzygies and Hilbert functions, Lect. Notes Pure Appl. Math. 254, 67-111 (2007), Chapman & Hall/CRC, Boca Raton, FL · Zbl 1127.13002 · doi:10.1201/9781420050912.ch3 |
| [9] | Cox, David; Goldman, Ronald; Zhang, Ming, On the validity of implicitization by moving quadrics of rational surfaces with no base points, J. Symbolic Comput., 29, 3, 419-440 (2000) · Zbl 0959.68124 · doi:10.1006/jsco.1999.0325 |
| [10] | Eisenbud, David, Commutative algebra, Graduate Texts in Mathematics 150, xvi+785 pp. (1995), Springer-Verlag, New York · Zbl 0819.13001 · doi:10.1007/978-1-4612-5350-1 |
| [11] | Harris, Joe, Algebraic geometry, Graduate Texts in Mathematics 133, xx+328 pp. (1992), Springer-Verlag, New York · Zbl 0779.14001 · doi:10.1007/978-1-4757-2189-8 |
| [12] | Hartshorne, Robin, Algebraic geometry, xvi+496 pp. (1977), Springer-Verlag, New York-Heidelberg · Zbl 0367.14001 |
| [13] | Herzog, J.; Simis, A.; Vasconcelos, W. V., Approximation complexes of blowing-up rings, J. Algebra, 74, 2, 466-493 (1982) · Zbl 0484.13006 · doi:10.1016/0021-8693(82)90034-5 |
| [14] | Herzog, J.; Simis, A.; Vasconcelos, W. V., Approximation complexes of blowing-up rings. II, J. Algebra, 82, 1, 53-83 (1983) · Zbl 0515.13018 · doi:10.1016/0021-8693(83)90173-4 |
| [15] | Schenck, Hal; Seceleanu, Alexandra; Validashti, Javid, Syzygies and singularities of tensor product surfaces of bidegree \((2,1)\), Math. Comp., 83, 287, 1337-1372 (2014) · Zbl 1286.13013 · doi:10.1090/S0025-5718-2013-02764-0 |
| [16] | [sc] T. W. Sederberg, F. Chen, Implicitization using moving curves and surfaces, in Proceedings of SIGGRAPH, 1995, 301-308. |
| [17] | Sederberg, Tom; Goldman, Ron; Du, Hang, Implicitizing rational curves by the method of moving algebraic curves, J. Symbolic Comput., 23, 2-3, 153-175 (1997) · Zbl 0872.68193 · doi:10.1006/jsco.1996.0081 |
| [18] | Sederberg, Thomas W.; Saito, Takafumi; Qi, Dong Xu; Klimaszewski, Krzysztof S., Curve implicitization using moving lines, Comput. Aided Geom. Design, 11, 6, 687-706 (1994) · Zbl 0875.68827 · doi:10.1016/0167-8396(94)90059-0 |
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.
