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Public viva-voce Notification Date: Thursday, 25th July 2024 Time: 3:30 - 5:00 PM Venue: Seminar Hall On $F$-rationality of blow-up algebras and Frobenius Betti numbers Nirmal Kotal Chennai Mathematical Institute. 25-07-24 Abstract In a commutative ring $R$ with a prime characteristic $p>0$, the Frobenius map $F: R \to R$, which sends $r$ to $r^p$, is crucial for analyzing singularities in positive characteristic. The $F$-rationality is a specific type of singularity in positive characteristic related to rational singularities in characteristic zero. My thesis addresses key questions in positive characteristic singularities, focusing on the $F$-rationality of blow-up algebras and the computation of Frobenius Betti numbers. In the first part of my talk, I will present some sufficient conditions for Cohen-Macaulay normal blow-up algebras to be $F$-rational. Our results provide conditions on the test ideals $\tau(I^n)$, $n \geq 1$, which imply that the normalization of the Rees algebra $R[It]$ is $F$-rational. Additionally, we show that if $R$ is a hypersurface of degree 2, or if the dimension is at most 3, the $F$-rationality of the blow-up algebra with respect to its maximal ideal is ensured by the $F$-rationality of its $\Proj$. Furthermore, I will describe the parameter test submodule of the blow-up algebras, which characterizes the non-$F$-rational locus (the set of prime ideals where the localization fails to be $F$-rational). The notion of Frobenius Betti numbers generalizes the Hilbert-Kunz multiplicity theory and serves as an invariant measuring singularities. This invariant can be estimated using the Betti numbers of sufficiently large twists induced by the Frobenius map. The second part of my talk will focus on explicit computations of these invariants for certain rings, particularly Cohen-Macaulay graded rings of finite Cohen-Macaulay type
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