## Tensor product surfaces and linear syzygies

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- by Eliana Duarte and Hal Schenck PDF
- Proc. Amer. Math. Soc.
**144**(2016), 65-72 Request permission

## 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.## References

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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
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 65-72 - MSC (2010): Primary 14M25; Secondary 14F17
- DOI: https://doi.org/10.1090/proc/12703
- MathSciNet review: 3415577