2010/07/20 by Botbol, Nicolás
#Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1007.3437
In this article we analyze the implicitization problem of the image of a rational map ϕ: X --> Pn, with T a toric variety of dimension n-1 defined by its Cox ring R. Let I:=(f0,...,fn) be n+1 homogeneous elements of R. We blow-up the base locus of ϕ, V(I), and we approximate the Rees algebra ReesR(I) of this blow-up by the symmetric algebra SymR(I). We provide under suitable assumptions, resolutions \Z. for SymR(I) graded by the torus-invariant divisor group of X, Cl(X), such that the determinant of a graded strand, det((\Z.)μ), gives a multiple of the implicit equation, for suitable μ∈ Cl(X). Indeed, we compute a region in Cl(X) which depends on the regularity of SymR(I) where to choose μ. We also give a geometrical interpretation of the possible other factors appearing in det((\Z.)μ). A very detailed description is given when X is a multiprojective space.