Research
My research has been mainly focused in Singularity Theory in Positive Characteristic, which lays in the field of Commutative Algebra. During my PhD I have been mostly thinking about singularities and numerical invariants derived from the Frobenius endomorphism, also known as F-singularities and F-invariants, with a focus on different aspects of the Watanabe–Yoshida conjecture, one of the main driving questions in the field of Hilbert–Kunz theory.
In this page, you can find a summary of some projects I have been working on. Feel free to reach out if you have any comments!
On the Watanabe–Yoshida conjecture.
See the arXiv preprint of this project.
Let \(R_{p,d} := \mathbb{F}_p[[x_0,\dots,x_d]]/(x_0^2+\dots+x_d^2)\) the ring of formal series at the origin of a simple double point of characteristic \(p\).
Conjecture (Watanabe–Yoshida, 2005). Let \((R,\mathfrak{m},k)\) be an unmixed nonregular local ring of characteristic \(p>0\). Then, \[e_\text{HK}(R)\geq e_\text{HK}(R_{p,d})\geq 1+c_d.\] Furthermore, if the first equality holds, then \(\widehat{R} \cong k[[x_0,\dots,x_d]]/(x_0^2+\dots+x_d^2)\).
On the Strong Watanabe–Yoshida conjecture
The Watanabe–Yoshida conjecture thus roughly states that the \(A_1\) singularity attains the infimum of the Hilbert–Kunz multiplicity, and is the only one to do so. This second part of the statement is what we call the "Strong" Watanabe–Yoshida conjecture. The goal of this project was settling the strong form of the conjecture for complete intersections, building on results by Enescu and Shimomoto. As a corollary, we also obtain the result in dimension up to 7 for general (unmixed) singularities:
Theorem (2025). Let \((R,\mathfrak{m},k)\) be an unmixed nonregular local ring of characteristic \(p>0\). Then, if the dimension of \(R\) is less than 7, or if \(R\) is a complete intersection, then \(e_\text{HK}(R)= e_\text{HK}(R_{p,d})\) implies that that \(\widehat{R} \cong k[[x_0,\dots,x_d]]/(x_0^2+\dots+x_d^2)\).
On the characteristic 2 case of the conjecture
Back in 2022, before I started my PhD at BCAM, I had the opportunity of doing an internship under the supervision of Ilya Smirnov. He suggested that we try to fill in some gaps in the literature of the Watanabe–Yoshida conjecture regarding characteristic 2.
The subtlety is that, in characteristic 2, the equation of an \(A_1\) changes slightly (see the classification of Simple Singularities in positive characteristic). If the dimension is odd, then the singularity is defined by \(x_0x_1+...+x_{d-1}x_{d}\), and if it is even, then by \(x_0x_1+...+x_{d-2}x_{d-1}+x_d^2\).
Theorem (2022). Let \((R,\mathfrak{m},k)\) be an unmixed nonregular local ring of characteristic 2. Then, if the dimension of \(R\) is less than 7, or if \(R\) is a complete intersection, then \(e_\text{HK}(R)\geq e_\text{HK}(R_{2,d})\), with equality only if \(\hat{R} = R_{2,d}\hat{\otimes} k\)