I am a PhD student in Mathematics at PUC-Rio working at the interface of algebraic geometry and mirror symmetry, especially on Fano and log del Pezzo surfaces. My recent work develops categorical aspects of their geometry through derived categories, exceptional collections, Brauer groups, and categorical invariants. I am currently also studying the quantum and enumerative side of mirror symmetry, particularly big quantum cohomology and Gromov–Witten theory for X10 ⊂ ℙ(1,2,3,5). Landau–Ginzburg models and periods provide a bridge between these categorical and enumerative directions.
Research advisor: Sergey Galkin, currently on sabbatical; co-advisor: Vasily Golyshev; official PhD advisor at PUC-Rio: Nicolau Saldanha.
I am currently studying big quantum cohomology and genus-zero Gromov–Witten theory for log del Pezzo surfaces, with X10 ⊂ ℙ(1,2,3,5) as a central testing ground. The aim is to understand how the enumerative and quantum side interacts with the mirror Landau–Ginzburg model, its periods, and the categorical structures appearing in my recent work.
My recent work studies derived categories, exceptional collections, Brauer groups, and categorical invariants of log del Pezzo surfaces with quotient singularities. The surface X10 is a recurring example linking explicit birational geometry to the categorical structure of its canonical stack and coarse space.
A parallel direction concerns explicit Landau–Ginzburg mirrors and their periods, including the seven-parameter mirror project for the cubic surface. This provides the mirror-theoretic side of the broader program connecting birational, categorical, and quantum information.
This paper computes the categorical genus of the canonical stack of a log del Pezzo surface with cyclic quotient singularities in terms of the local widths and Gorenstein indices of the singularities. It relates the invariant to age-less-than-one inertia, tracks its behavior under sufficiently general Q-Gorenstein deformations, and identifies the residual invariant with Tveiten’s mutable genus for surfaces admitting toric Q-Gorenstein degenerations. For toric surfaces, the categorical genus of the canonical stack counts all interior lattice points and agrees with the genus of a general Laurent-polynomial fiber.
We study the derived categories of log del Pezzo surfaces with only 1/3(1,1) singularities. For the Fano index one families classified by Corti and Heuberger, we construct explicit full exceptional collections on their canonical stacks using blow-up presentations of the minimal resolutions and cascades of smooth points. More generally, for any log del Pezzo surface X with k such singularities, we obtain the length formula 12 - KX2 + 4k/3, independent of the Fano index. We show that the intersection lattice of the exceptional curves determines the Brauer group: it vanishes for k ≤ 5 and is isomorphic to Z/3 for k = 6. Applying the Karmazyn–Kuznetsov–Shinder framework, we obtain semiorthogonal decompositions of the bounded derived categories of coherent sheaves on the singular surfaces for the one- and three-point cascades, together with a nontrivial Brauer-twisted decomposition for the six-point cascade. For quasismooth degree-10 hypersurfaces in ℙ(1,2,3,5), we construct a thirteen-object full exceptional collection on the canonical stack, compute its Euler matrix, and obtain a semiorthogonal decomposition of the derived category of the singular surface. We also identify a general such hypersurface as the anticanonical model of the blow-up of ℙ(1,1,3) at eight general smooth points, with the converse holding for general centers. Finally, for rational surfaces with arbitrary baskets of 1/ni(1,1) singularities, we give a general block formula for the Euler matrices of full exceptional collections on their canonical stacks.
Certificate summary, checksums, source code for the computations, and retained logs are organized in the repository.
I was awarded the FAPERJ Nota 10 Excellence Scholarship for my PhD thesis project, Functional Equations for Landau–Ginzburg Models and Exponential Motives . The project studies graph potentials, exponential integrals and sums, Kloosterman-type phenomena, trace formulas, and arithmetic realizations of mirror-symmetric structures. This project provided the Landau–Ginzburg and period-theoretic foundation for my current work on categorical and quantum aspects of Fano and log del Pezzo geometry.
Spanish native; English advanced; Portuguese advanced; basic Chinese and French.
A small personal note: one song currently on my playlist.
Email:
alex.gomez@pucp.edu.pe
GitHub:
github.com/alequisGS
CV:
AlexGomez-CV.pdf
Thesis project:
FAPERJ Nota 10 project PDF
