A formula for the *-core of an ideal

Expanding on the work of Fouli and Vassilev \cite{FV}, we determine a formula for the *-$\rm{core}$ of an ideal in two different settings: (1) in a Cohen--Macaulay local ring of characteristic $p>0$, perfect residue field and test ideal of depth at least two, where the ideal has a minimal *-reduction that is a parameter ideal and (2) in a normal local domain of characteristic $p>0$, perfect residue field and \m-primary test ideal, where the ideal is a sufficiently high Frobenius power of an ideal. We also exhibit some examples where our formula fails if our hypotheses are not met.Comment: 11 pages, submitted for publicatio

Looking out for stable syzygy bundles

We study (slope-)stability properties of syzygy bundles on a projective space P^N given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring.Comment: This paper contains an appendix by Georg Hein: Semistability of the general syzygy bundle. The new version is quite ne