Bigraded structures and the depth of blow-up algebras

maximal depth property of bigraded modules

Let $S=K[x_1, \dots, x_m, y_1, \dots, y_n]$ be the standard bigraded polynomial ring over a field $K$. Let $M$ be a finitely generated bigraded $S$-module and $Q=(y_1, \dots, y_n)$. We say $M$ has maximal depth with respect to $Q$ if there is an associated prime $\pp$ of $M$ such that $\grade(Q, M)=\cd(Q, S/\pp)$. In this paper, we study finitely generated bigraded modules with maximal depth with respect to $Q$. It is shown that sequentially Cohen--Macaulay modules with respect to $Q$ have maximal depth with respect to $Q$. In fact, maximal depth property generalizes the concept of sequentiall…

On the diagonals of a Rees algebra

The aim of this work is to study the ring-theoretic properties of the diagonals of a Rees algebra, which from a geometric point of view are the homogeneous coordinate rings of embeddings of blow-ups of projective varieties along a subvariety. In Chapter 1 we extend the definitions and results about the biprojective scheme and the Hilbert polynomial of finitely generated bigraded modules defined over standard bigraded k-algebras to finitely generated bigraded S-modules, for S=k[X_1,..., X_n,Y_1,...,Y_r] the polynomial ring bigraded by deg X_i=(1,0), deg Y_j=(d_j,1), d_j \geq 0. We also relate t…

Multigraded rings, diagonal subalgebras, and rational singularities

We study the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise naturally as homogeneous coordinate rings of blow-ups of projective varieties. As a consequence of some of the results obtained here, it is shown that there exist standard bigraded hypersurfaces whose rings of invariants under torus actions have rational singularities, but are not of F-regular type. Another application is the construction of families of rings wi…

On the depth of blow-up rings of ideals of minimal mixed multiplicity

We show that if $(R, \m)$ is a Cohen-Macaulay local ring and $I$ is an ideal of minimal mixed multiplicity, then $\depth G(I) \geq d- 1$ implies that $\depth F(I) \geq d-1$. We use this to show that if $I$ is a contracted ideal in a two dimensional regular local ring then $\depth R[It]-1= \depth G(I) = \depth F(I)$. We also give an infinite class of ideals where $R[It]$ is Cohen-Macaulay but $F(I)$ is not.

Positivity of mixed multiplicities

This paper studies mixed multiplicities of an arbitrary standard bigraded algebra and mixed multiplicities of two ideals I, J in a local ring (A,m), where I is an m-primary ideal and J an arbitrary ideal. The main results are criteria for their positivity which can be used to compute them effectively. We also show that the range of positive mixed multiplicities of a bigraded algebra is rigid if the algebra satisfies the first chain condition and is connected in codimension one and that this range is always rigid for mixed multiplicities of ideals. These results can be used to study the mu-inva…

哪些级数是多项式环上分次模的Hilbert级数？

Which series are Hilbert series of graded modules over polynomial rings?

该研究刻画了多重分次多项式环上有限生成模的Hilbert级数对应的形式Laurent级数，给出了给定深度模的Hilbert级数的必要条件，并完全分类了双分次下正深度有限生成模的Hilbert级数。