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 Free Access
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.
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Additional Information
Hal Schenck
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
MR Author ID:
621581
Email:
schenck@math.uiuc.edu
Alexandra Seceleanu
Affiliation:
Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588
MR Author ID:
896988
ORCID:
0000-0002-7929-5424
Email:
aseceleanu2@math.unl.edu
Javid Validashti
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801
Email:
jvalidas@illinois.edu
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.