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Mathematics of Computation

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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
Published electronically: August 14, 2013
MathSciNet review: 3167461
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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.

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Additional Information

Hal Schenck
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
MR Author ID: 621581

Alexandra Seceleanu
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
MR Author ID: 896988
ORCID: 0000-0002-7929-5424

Javid Validashti
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

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.