Research & Projects

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.

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

On a Grauert-Riemenschneider vanishing theorem in dimension 3

Suppose (R, m) is an excellent local ring of dimension 3 and has rational singularities. Let π : X −→ Spec R be a blow-up and ϕ : W −→ X be any projective, birational morphism such that X and W are both normal, Cohen-Macaulay and have pseudorational singularitities in codimension 2. Then R^iϕ_∗ω_W = 0 and R^iπ_∗ω_X = 0 for all i > 0 and X has rational singularities. We use this result to prove Lipman’s vanishing conjecture in dimension 3 for arbitrary characteristics and provide a few applications.