The implicit equation of a multigraded hypersurface

Abstract

In this article we analyze the implicitization problem of the image of a rational map φ : X P n , with X a toric variety of dimension n − 1 defined by its Cox ring R. Let I := (f 0 , . . . , f n ) be n + 1 homogeneous elements of R. We blow-up the base locus of φ, 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 • ) 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 Sym R (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.

Implicitization of rational hypersurfaces via linear syzygies: A practical overview

“…Taking (ν 1 , ν 2 ) = (2a − 1, b − 1) one has that in STEP 1 a basis of syzygies (h Equality (3.2) also follows from Theorem 5.5 in [Bot11b].…”

“…Goldman et al studied the cases of planar and space curves [JG09,HWJG10,JWG10]. A generalization of the linear syzygy method when the support of the input polynomials is a square (that is, bihomogeneous of degree (d, d)) was proposed by Busé and Dohm [BD07], and for any polygon by Botbol, Dickenstein and Dohm [BDD09], and Botbol [Bot09,Bot11a,Bot11b]. This method is particularly adapted when the polynomials defining the parametrization are sparse, which is often the case.…”

Implicitization of tensor product surfaces in the presence of a generic set of basepoints

“…Following Theorem 5.1, ν = (5, 0). The ideal I U has two syzygies in degree (2, 0) and three syzygies in bidegree (1,1). The syzygies in degree (2, 0) span a free R-module and determine a basis for the syzygies of bidegree (5, 0) by multiplying by {s 3 , s 2 t, st 2 , t 3 } .…”

“…Proof. From the proof of Lemma 7.3 in Botbol [1], we know that Z ν is acyclic and that (Z 3 ) ν = 0. Hence Z ν has the form…”

“…An introduction to the use of these complexes to obtain implicit equations of parameterized hypersurfaces and the connection with Rees algebras can be found in Chardin's [5] work. These techniques were generalized for multigraded hypersurfaces such as tensor product surfaces by Botbol [1]. In Section 5 we use Botbol's results to obtain the implicit equation of X U via Z.…”