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Name: SATHER-WAGSTAFF, Sean
Title: Intersections of Symbolic Powers of Prime Ideals
Abstract: Let $(R,M)$ be a regular local ring with prime ideals $P$ and
$Q$ such that the ideal $P+Q$ is $M$-primary and
$\dim(R/P)+\dim(R/Q)=\dim(R)$. If $R$ contains a field, then we prove
that $P^{(r)}\cap Q^{(s)}\subseteq M^{r+s}$ for all positive integers $r$
and $s$, where $P^{(r)}$ is the $r$th symbolic power of $P$ and similarly
for $Q^{(s)}$. We conjecture
that this result holds when $R$ is a regular local ring of
mixed characteristic. We discuss the
background and motivation for this result, especially noting the work of
Kurano and Roberts on Serre's Positivity Conjecture.
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