Let $k$ be a field of characteristic $0,\,R\,=\,k\left[ {{x}_{1}},\,\ldots ,\,{{x}_{d}} \right]$ be a polynomial ring, and $m$ its maximal homogeneous ideal. Let $I\,\subset \,R$ be a homogeneous ideal in $R$. Let $\lambda (M)$ denote the length of an $R$-module $M$. In this paper, we show that

$$\underset{x\to \infty }{\mathop{\lim }}\,\,\frac{\lambda \left( H_{m}^{0}\left( R/{{I}^{n}} \right) \right)}{{{n}^{d}}}\,=\,\underset{x\to \infty }{\mathop{\lim }}\,\,\frac{\lambda \left( \text{Ext}_{R}^{d}\left( R/{{I}^{n}},\,R\left( -d \right) \right) \right)}{{{n}^{d}}}$$

always exists. This limit has been shown to be $e(I)/d!$ for $m$-primary ideals $I$ in a local Cohen–Macaulay ring, where $e(I)$ denotes the multiplicity of $I$. But we find that this limit may not be rational in general. We give an example for which the limit is an irrational number thereby showing that the lengths of these extension modules may not have polynomial growth.

We study the Hilbert functions of fat points in ${{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$. If $Z\,\subseteq \,{{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ is an arbitrary fat point scheme, then it can be shown that for every $i$ and $j$ the values of the Hilbert function ${{H}_{Z}}(l,\,j)$ and ${{H}_{Z}}(i,\,l)$ eventually become constant for $l\,\gg \,0$. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ${{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$. This enables us to compute all but a finite number values of ${{H}_{Z}}$ without using the coordinates of points. We also characterize the $\text{ACM}$ fat point schemes using our description of the eventual behaviour. In fact, in the case that $Z\,\subseteq \,{{\mathbb{P}}^{1\,}}\times \,{{\mathbb{P}}^{1}}$ is $\text{ACM}$, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

Let (R, ${\frak {m}}$) be a Cohen–Macaulay local ring and let I be an ideal. There are at least five algebras built on I whose multiplicity data affect the reduction number r(I) of the ideal. We introduce techniques from the Rees algebra theory of modules to produce estimates for r(I), for classes of ideals of dimension one and two. Previous cases of such estimates were derived for ideals of dimension zero.

We present an elementary proof that given a general collection of d points in Pn the linear system of cubics singular on each point has the expected codimension except when n=4 and d=7. In that case the cubic is unique. This, together with previous work of the author, gives a proof of the Alexander–Hirschowitz interpolation theorem.

In an earlier work, the authors described a mechanism for lifting monomial ideals to reduced unions of linear varieties. When the monomial ideal is Cohen–Macaulay (including Artinian), the corresponding union of linear varieties is arithmetically Cohen–Macaulay. The first main result of this paper is that if the monomial ideal is Artinian then the corresponding union is in the Gorenstein linkage class of a complete intersection (glicci). This technique has some interesting consequences. For instance, given any (d + 1)-times differentiable O-sequence H, there is a nondegenerate arithmetically Cohen–Macaulay reduced union of linear varieties with Hilbert function H which is glicci. In other words, any Hilbert function that occurs for arithmetically Cohen–Macaulay schemes in fact occurs among the glicci schemes. This is not true for licci schemes. Modifying our technique, the second main result is that any Cohen–Macaulay Borel-fixed monomial ideal is glicci. As a consequence, all arithmetically Cohen–Macaulay subschemes of projective space are glicci up to flat deformation.

Let R be a homogeneous ring over an infinite field, I⊂R a homogeneous ideal, and $\frak a$⊂I an ideal generated by s forms of degrees d1,…,ds so that codim($\frak a$:I)[ges ]s. We give broad conditions for when the Hilbert function of R/$\frak a$ or of R/($\frak a$:I) is determined by I and the degrees d1,…,ds. These conditions are expressed in terms of residual intersections of I, culminating in the notion of residually S2 ideals. We prove that the residually S2 property is implied by the vanishing of certain Ext modules and deduce that generic projections tend to produce ideals with this property.