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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/proc/12703
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
Email: emduart2@math.uiuc.edu

Hal Schenck
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
MR Author ID: 621581
Email: schenck@math.uiuc.edu

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