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Full-Text Articles in Applied Mathematics
Symbolic Rees Algebras, Eloísa Grifo, Alexandra Seceleanu
Symbolic Rees Algebras, Eloísa Grifo, Alexandra Seceleanu
Department of Mathematics: Faculty Publications
We survey old and new approaches to the study of symbolic powers of ideals. Our focus is on the symbolic Rees algebra of an ideal, viewed both as a tool to investigate its symbolic powers and as a source of challenging problems in its own right. We provide an invitation to this area of investigation by stating several open questions.
Expected Resurgence Of Ideals Defining Gorenstein Rings, Eloísa Grifo, Craig Huneke, Vivek Mukundan
Expected Resurgence Of Ideals Defining Gorenstein Rings, Eloísa Grifo, Craig Huneke, Vivek Mukundan
Department of Mathematics: Faculty Publications
Building on previous work by the same authors, we show that certain ideals defining Gorenstein rings have expected resurgence, and thus satisfy the stable Harbourne Conjecture. In prime characteristic, we can take any radical ideal defining a Gorenstein ring in a regular ring, provided its symbolic powers are given by saturations with the maximal ideal. While this property is not suitable for reduction to characteristic p, we show that a similar result holds in equicharacteristic 0 under the additional hypothesis that the symbolic Rees algebra of I is noetherian.
Are Symbolic Powers Highly Evolved?, Brian Harbourne, Craig Hunkeke
Are Symbolic Powers Highly Evolved?, Brian Harbourne, Craig Hunkeke
Department of Mathematics: Faculty Publications
Searching for structural reasons behind old results and conjectures of Chudnovksy regarding the least degree of a nonzero form in an ideal of fat points in PN, we make conjectures which explain them, and we prove the conjectures in certain cases, including the case of general points in P2. Our conjectures were also partly motivated by the Eisenbud-Mazur Conjecture on evolutions, which concerns symbolic squares of prime ideals in local rings, but in contrast we consider higher symbolic powers of homogeneous ideals in polynomial rings.