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We study the base locus of the higher fundamental forms of a projective toric variety $X$ at a general point. More precisely we consider the closure $X$ of the image of a map $({\mathbb C}^*)^k\to {\mathbb P}^n$, sending $t$ to the vector of Laurent monomials with exponents $p_0,\dots,p_n\in {\mathbb Z}^k$. We prove that the $m$-th fundamental form of such an $X$ at a general point has non empty base locus if and only if the points $p_i$ lie on a suitable degree-$m$ affine hypersurface. We then restrict to the case in which the points $p_i$ are all the lattice points of a lattice polytope and we give some applications of the above result. In particular we provide a classification for the second fundamental forms on toric surfaces, and we also give some new examples of weighted $3$-dimensional projective spaces whose blowing up at a general point is not Mori dream., Comment: Final version, to appear on JPAA

The effective relations and the relation type of an ideal generated by two regular elements can be expressed in terms of the ring of global sections of the blowing-up along the ideal. As a consequence, we find the minimal overring where the ideal is syzygetic or of linear type.

Let k be a field. Spivakovsky's theorem on the solution of Hironaka's polyhedral game has been extended by Bloch to show that a morphism f : Z→S of finite type k-schemes can be put in good position with respect to a normal crossing divisor ∂S on S by taking the proper transform with respect to an iterated blowing up of faces of ∂S. We extend these results to schemes of finite type over a regular scheme of dimension one, including the case of mixed characteristic.

For certain classes of rings we give an affirmative answer to whether there exists a uniform bound on the least power of the tight closure of an arbitrary ideal which lies in the ideal. The rings need to have the property that modulo each minimal prime there exists a resolution of singularities obtained by blowing up an ideal. In this case we prove the existence of a ‘strong’ test ideal, and then apply this existence to give an affirmative answer to the uniform bound question.

Let G(r,n) be the Grassmannian of r-dimensional subspaces of Kn. With each sequence t = (t1,…,tr) of positive integers such that 1

Let ( R , m) be a one-dimensional Gorenstein local ring and I an m-primary ideal of R . In this paper, generalizing a result by Orecchia and Ramella (Manuscripta Math. 68 (1990) 1-7), we give necessary and sufficient conditions for the conductor of the blowing-up of R along I being equal to a power of I in terms of the reduction exponent and the genus of I .

Let (R,m) be a 2-dimensional rational singularity with algebraically closed residue field and for which the associated graded ring is an integrally closed domain. According to Göhner, (R,m) satisfies condition (N): given a prime divisor v, there exists a unique complete m-primary ideal Av in R with T(Av)={v} and such that any complete m-primary ideal with unique Rees valuation v, is a power of Av. We use the theory of degree functions developed by Rees and Sharp as well as some results about regular local rings, to investigate the degree coefficients d(Av,v). As an immediate corollary, we find that for a simple complete m1-primary ideal I1 in an immediate quadratic transform (R1,m1) of (R,m); the inverse transform of I1 in R is projectively full.

Given a birational projective morphism of quasi-projective varieties f : Z→X . We want to find the ideal sheaf I over X such that the blowing up of X along I corresponds to f . In this paper we approach the problem from two directions, solving two subcases. First we present a method that determines I when f is the composition of blowing ups along known centers, then by another method, we compute I directly from i(Z)⊂ P X n .