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

Hal Schenck, Alexandra Seceleanu, Javid Validashti

Research output: Contribution to journalArticlepeer-review


Let U ⊆ H0(O ℙ1×ℙ1 (2, 1)) be a basepoint free four-dimensional vector space. The sections corresponding to U determine a regular map φU : ℙ1 × ℙ1 -→ ℙ3. We study the associated bigraded ideal IU ⊆ 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(ℙ1 × ℙ1), via work of Busé-Jouanolou, Busé-Chardin, Botbol and Botbol-Dickenstein-Dohm on the approximation complex Z. In four of the six cases IU 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 languageEnglish (US)
Pages (from-to)1337-1372
Number of pages36
JournalMathematics of Computation
Issue number287
StatePublished - May 2014
Externally publishedYes


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

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Computational Mathematics
  • Applied Mathematics


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