Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 

 

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
DOI: https://doi.org/10.1090/S0025-5718-2013-02764-0
Published electronically: August 14, 2013
MathSciNet review: 3167461
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 14M25, 14F17

Retrieve articles in all journals with MSC (2010): 14M25, 14F17


Additional Information

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

Alexandra Seceleanu
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
Email: aseceleanu2@math.unl.edu

Javid Validashti
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email: jvalidas@illinois.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02764-0
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.