The implicit equation of a multigraded hypersurface

JOURNAL OF ALGEBRA

卷:348

Article

Botbol, Nicolas^1,2 

[1] Univ Buenos Aires, FCEN, Dept Matemat, RA-1053 Buenos Aires, DF, Argentina

[2] Univ P&M Curie, Inst Math Jussieu, Paris VI, France

关键词: Implicitization;    Implicit equation;    Hypersurfaces;    Toric variety;    Elimination theory;    Koszul complex;    Approximation complex;    Resultant;    Graded ring;    Multigraded ring;    Graded algebra;    Multigraded algebra;    Castelnuovo-Mumford regularity;   

DOI  :  10.1016/j.jalgebra.2011.09.019

来源: Elsevier

【 摘 要 】

In this article we analyze the implicitization problem of the image of a rational map phi : X -> P(n), with X a toric variety of dimension n - 1 defined by its Cox ring R. Let I := (integral(0,) ... integral(n)) be n + 1 homogeneous elements of R. We blow-up the base locus of phi, V(I), and we approximate the Rees algebra Rees(R)(I) of this blow-up by the symmetric algebra Sym(R) (I). We provide under suitable assumptions, resolutions Z. for Sym(R)(I) graded by the divisor group of X. Cl(X). such that the determinant of a graded strand, det((Z.)mu). gives a multiple of the implicit equation, for suitable mu is an element of Cl. Indeed, we compute a region in Cl(X) which depends on the regularity of Sym(R)(I) where to choose mu. We also give a geometrical interpretation of the possible other factors appearing in det((Z.)mu). A very detailed description is given when X is a multiprojective space. (C) 2011 Elsevier Inc. All rights reserved.

【 授权许可】

Free