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 Skoda-type applications.

Paper (in preparation)

SESHADRI CONSTANTS IN MIXED CHARACTERISTICS

(Joint with Joe Waldron)

We prove various natural properties (similar to that of equi-characteristic 0 and p) of semiample Seshadri constants (as defined by Bhatt et al.) in mixed characteristic. Then we prove a characterization of projective space in mixed characteristic, again similar to that of equi-characteristic 0 and p. In the appendix, by an example constructed by A. Langer, we show that generic lower bound (as proved by Ein, K\"uchle and Lazarsfeld in characteristic 0) does not hold in char p>0, for any prime p and in any dimension >1, thereby answering a question of Patakfalvi, Schwede, Tucker and others negatively.

Paper (in preparation)

KOV ´ACS VANISHING FOR EXCELLENT LOG CANONICAL

THREEFOLDS

(Joint with Shikha Bhutani)

We prove a vanishing theorem of S. Kov´acs for excellent log canonical threefolds whose closed points have perfect residue fields of characteristic p > p_0. This vanishing theorem roughly says that LC threefold pairs are not far from being rational pairs (in the sense of Koll´ar & Kov´acs and Schwede & Takagi) if one includes a correction term. We apply this vanishing to prove a depth relation similar to Chou’s theorem and resolution-invariance of cohomologies of “essential divisors” for isolated log canonical threefolds in large characteristic similar to Ishii’s and Fujino’s results in char 0. Using examples constructed by Bernasconi and Cascini-Tanaka, we show that these results fail in low characteristic. Finally, we show Kov´acs and Alexeev-Hacon’s result (in char 0) that “Irrational centers are LC centers” fails in every positive characteristic.