Tensor product surfaces and linear syzygies

Eliana Duarte, Hal Schenck

Research output: Contribution to journalArticlepeer-review


Let U ⊆ H0(O1×ℙ1 (a, b)) be a basepoint free four-dimensional vector space, with a, b ≥ 2. The sections corresponding to U determine a regular map ϕU : ℙ1 × ℙ1 → ℙ3. We show that there can be at most one linear syzygy on the associated bigraded ideal IU ⊆ k[s, t; u, v]. Existence of a linear syzygy, coupled with the assumption that U is basepoint free, implies the existence of an additional “special pair” of minimal first syzygies. Using results of Botbol, we show that these three syzygies are sufficient to determine the implicit equation of ϕU (ℙ1 × ℙ1), and that Sing(ϕU (ℙ1 × ℙ1)) contains a line.

Original languageEnglish (US)
Pages (from-to)65-72
Number of pages8
JournalProceedings of the American Mathematical Society
Issue number1
StatePublished - Jan 2016


  • Bihomogeneous ideal
  • Syzygy
  • Tensor product surface

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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