## Abstract

Let U ⊆ H^{0}(O _{ℙ1×ℙ1} (2, 1)) be a basepoint free four-dimensional vector space. The sections corresponding to U determine a regular map φU : ℙ^{1} × ℙ^{1} -→ ℙ^{3}. We study the associated bigraded ideal I_{U} ⊆ 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 φU(ℙ^{1} × ℙ^{1}), via work of Busé-Jouanolou, Busé-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex 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.

Original language | English (US) |
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Pages (from-to) | 1337-1372 |

Number of pages | 36 |

Journal | Mathematics of Computation |

Volume | 83 |

Issue number | 287 |

DOIs | |

State | Published - May 2014 |

Externally published | Yes |

## Keywords

- Bihomogeneous ideal
- Segre-Veronese map
- Tensor product surface

## ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics