2008/03/05 by Nicolás Botbol, Botbol, Nicolas
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · doi:10.48550/arxiv.0803.0573
openalex publication_date 2008/03/05 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28
We develop in this paper some methods for studying the implicitization problem for a rational map ϕ: ℙn → (ℙ1)n+1 defining a hypersurface in (ℙ1)n+1, based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay Resultants and Koszul complexes coincides, in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions. Under suitable hypotheses, these techniques enable us to obtain the implicit equation, up to a power, and up to some other extra factor. We give algebraic and geometric conditions for determining when the computed equation defines the scheme theoretic image of ϕ, and, what are the extra varieties that appear. We also give some applications to the problem of computing sparse discriminants.