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Tensor product surfaces and linear syzygies

Authors: Eliana Duarte and Hal Schenck
Journal: Proc. Amer. Math. Soc. 144 (2016), 65-72
MSC (2010): Primary 14M25; Secondary 14F17
Published electronically: June 9, 2015
MathSciNet review: 3415577
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Abstract: 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 \ge 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.

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Additional Information

Eliana Duarte
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Hal Schenck
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Keywords: Tensor product surface, bihomogeneous ideal, syzygy
Received by editor(s): August 4, 2014
Received by editor(s) in revised form: December 23, 2014
Published electronically: June 9, 2015
Additional Notes: The second author was supported by NSF 1068754, NSF 1312071
Communicated by: Irena Peeva
Article copyright: © Copyright 2015 American Mathematical Society

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