Tensor product surfaces and linear syzygies

2014/02/27 by Eliana Duarte, Duarte, Eliana, Hal Schenck +1
Computer Science · Mathematics · #14F17 #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Numerical Analysis (math.NA) #Polynomial and algebraic computation #Primary 65D17 #Secondary 14M25

paper · pdf · doi:10.48550/arxiv.1402.6751

openalex publication_date 2014/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let U be a basepoint free four-dimensional subpace of the space of sections of bidegree (a,b) on X = P¹ x P¹, with a and b at least 2. The sections corresponding to U determine a regular map from X to P³. We show that there can be at most one linear syzygy on the associated bigraded ideal I_U in the bigraded ring 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 the image of X in P³; we also show that the singular locus must contain a line.

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