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Research Interests
Singularities arising in the Minimal Model Program.
Singularities defined by the Frobenius map.
Vanishing theorems in Birational Algebraic Geometry.
Bertini-type theorems.
Seshadri constants.
Given a reduced, local ring R and an ideal a of positive height, we give a decomposition of the test module of the extended Rees algebra.
In particular, the degree zero component of this test module is the test module of (R, a), thereby reducing the computation of test modules for non-principal ideals to the much easier case of principal ideals. Additionally, we apply our decomposition to generalize results on the F-rationality of Rees and extended Rees algebras, as well as give simplified proofs of the discreteness and rationality of F-jumping numbers, among other applications. We also prove a cool result relating canonical modules of Rees and extended Rees algebras.
Multiplier Modules of extended Rees algebras
Multiplier Modules of extended Rees algebras
Here, we prove exactly similar decompositions for multiplier modules of extended Rees algebras in characteristic 0.
However, the proof is very different from the positive characteristics case: here, we use the geometry of deformation to the normal cone and toric computations!
Paper
Paper
(Under preparation)
A Short Note on Grauert-Riemenschneider Vanishing Theorem and Rational Singularities
A Short Note on Grauert-Riemenschneider Vanishing Theorem and Rational Singularities
Ongoing Project
Ongoing Project
F-jumping numbers of non-Q-Gorenstein varieties
F-jumping numbers of non-Q-Gorenstein varieties
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