Constraints for the normality of monomial subrings and birationality

Abstract

Abstract. Let k be a field and let F ⊂ k[x 1 , . . . , xn] be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra R [Ft] and the subring k [F]. If the monomials in F have the same degree, one of the consequences is a criterion for the k-rational map F :defined by F to be birational onto its image.

Degree of Rational Maps versus Syzygies

“…One may notice that Lemma 2.2, item (3), and Theorem 3.3 give an improved upper bound, k defined by the monomials {x 2 , yz, z 2 }. For maps defined by monomials, [SiVi,Proposition 2.1] gives an easy criterion of birationality in terms of the gcd of the maximal minors of the corresponding log-matrix, namely, that the latter has to be ±d, where d is the common degree of the monomials. In this case it is nearly trivial to see that Φ is not birational because the matrix is square of determinant 4.…”

Counting Square free Cremona monomial maps

“…The following definition has an algebro-geometric flavor, including an algebraic notion of birationality. It is very useful in this context, for more details see [SV1,SV2,CS]. An ordered set F of n monomials of same degree…”

“…Proposition 2.6. [SV1] (Determinantal Principle of Birationality (DPB)) Let F be a finite set of monomials of the same degree d and let A F be its log matrix. Then F is a Cremona set if and only if det A F = ±d.…”

“…Cremona transformations are birational automorphism of the projective space, they were firstly systematically studied by L. Cremona in the 19th century and remains a major classical topic in Algebraic Geometry. The interest in monomial Cremona transformations, otherwise has flourished more recently as one can see, for example in [V,GP,SV1,SV2,SV3]. Following the philosophy introduced in [SV1,SV2] and shared by [SV3,CS] we study the so called "birational combinatorics" meaning, see loc.…”

“…The interest in monomial Cremona transformations, otherwise has flourished more recently as one can see, for example in [V,GP,SV1,SV2,SV3]. Following the philosophy introduced in [SV1,SV2] and shared by [SV3,CS] we study the so called "birational combinatorics" meaning, see loc. cit., the theory of characteristic-free rational maps of the projective space defined by monomials, along with natural criteria for such maps to be birational.…”