F-thresholds and Test ideals for determinantal ideals

Ines Henriques (University of Sheffield)

Wednesday 12th February, 2014 16:00-17:00 Maths 204


Test ideals first appeared in the theory of tight closure, and reflect the singularities of a ring of positive characteristic. Motivated by their close connection to multiplier ideals in characteristic zero, N. Hara and K. Yoshida defined generalized test ideals as their characteristic p analogue. Whereas multiplier ideals are defined geometrically, using log resolutions, or even analytically, using integration, test ideals are defined algebraically using the Frobenius morphism. The generalized test ideals of an ideal I form a non-increasing, right continuous family, {τ(c . I)}, parametrized by a positive real parameter c. The points of discontinuity in this parametrization, are called F-thresholds of I and form a discrete subset of the rational numbers (work of Blickle-Mustaţă-Smith, Hara, Takagi-Takahashi, Schwede-Takagi, Katzman-Lyubeznik-Zhang,Schwede-Tucker-Zhang). We consider ideals generated by minors (of maximal size) of a matrix of indeterminates, in its polynomial ring over a field of positive characteristic. Using a purely algebraic approach, we give a complete description of their F-thresholds and generalized test ideals.

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