Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals

Let $I$ be an ideal whose symbolic Rees algebra is Noetherian. For $m \geq 1$, the $m$-th symbolic defect, sdefect$(I,m)$, of $I$ is defined to be the minimal number of generators of the module $\frac{I^{(m)}}{I^m}$. We prove that sdefect$(I,m)$ is eventually quasi-polynomial as a function in $m$. We compute the symbolic defect explicitly for certain monomial ideals arising from graphs, termed cover ideals. We go on to give a formula for the Waldschmidt constant, an asymptotic invariant measuring the growth of the degrees of generators of symbolic powers, for ideals whose symbolic Rees algebra is Noetherian.Comment: 22 pages, 5 figure

On quasi-equigenerated and Freiman cover ideals of graphs

A quasi-equigenerated monomial ideal $I$ in the polynomial ring $R= k[x_1, \ldots, x_n]$ is a Freiman ideal if $\mu(I^2) = l(I)\mu(I)- \binom{l(I)}{2}$ where $l(I)$ is the analytic spread of $I$ and $\mu(I)$ is the number of minimal generators of $I$. Freiman ideals are special since there exists an exact formula computing the minimal number of generators of any of their powers. In this work we address the question of characterizing which cover ideals of simple graphs are Freiman.Comment: 26 page

Some Analytic generalizations of the Briancon-Skoda Theorem

<p>The Brian\c con-Skoda theorem appears in many variations in recent literature. The common denominator is that the theorem gives a sufficient condition that implies a membership \phi\in \ideala^l, where \ideala is an ideal of some ring $R$. In the analytic interpretation $R$ is the local ring of an analytic space $Z$, and the condition is that |\phi|\leq C|\ideala|^{N+l} holds on the space $Z$. The theorem thus relates the rate of vanishing of $\phi$ along the locus of \ideala to actual membership of (powers of) the ideal. The smallest integer $N$ that works for all \ideala \subset R and all $l\geq 1$ simultaneously will be called the Brian\c con-Skoda number of the ring $R$. </p> <p>The thesis contains three papers. The first one gives an elementary proof of the original Brian\c con-Skoda theorem. This case is simply Z=\C^n.</p> <p>The second paper contains an analytic proof of a generalization by Huneke. The result is also sharper when \ideala has few generators if the geometry is not to complicated in a certain sense. Moreover, the method can give upper bounds for the Brian\c con-Skoda number for some varieties such as for example the cusp $z^p = w^q$.</p> <p>In the third paper non-reduced analytic spaces are considered. In this setting Huneke's generalization must be modified to remain valid. More precisely, $\phi$ belongs to \ideala^l if one requires that |L \phi| \leq C |\ideala|^{N+l} holds on $Z$ for a given family of holomorphic differential operators on $Z$. We impose the assumption that the local ring \O_Z is Cohen-Macaulay for technical reasons.</p