Syzygies and singularities of tensor product surfaces of bidegree (2,1)

Schenck, Hal; Seceleanu, Alexandra; Validashti, Javid

doi:10.1090/S0025-5718-2013-02764-0

Syzygies and singularities of tensor product surfaces of bidegree $(2,1)$
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by Hal Schenck, Alexandra Seceleanu and Javid Validashti PDF

Math. Comp. 83 (2014), 1337-1372 Request permission

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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