Assistant Professor at the Department of Mathematics of the Universidad Técnica Federico Santa María, Valparaíso, Chile.
From September 2 to November 29, 2024, I will be a Visiting Researcher ('Poste Rouge') funded by the Institute of Mathematics of the CNRS (INSMI) at the LMA of the University of Poitiers.
Administrative services
Since 2024, I am the Vice-President of the Chilean Mathematical Society (SOMACHI).
Research
I am mainly interested in birational geometry. More precisely, in asymptotic invariants and positivity in algebraic geometry (such as linear series, Newton-Okounkov bodies, etc) and the use of techniques from the Minimal Model Program to the classification of algebraic varieties (e.g. Fano varieties). I am also interested in vector bundles and algebraic group actions on algebraic varieties.
(Pre)publications
Here you can find my research articles.
► "On singular Fano varieties with a divisor of Picard number one" (arXiv:1605.08307)
Annali della Scuola Normale Superiore di Pisa (5), XIX (2), pages 567–600, 2019.
DOI: 10.2422/2036-2145.201609_018.
► "Newton-Okounkov bodies on projective bundles over curves" (arXiv:1703.06828)
Mathematische Zeitschrift, 291 (3–4), pages 1357–1379, 2019.
DOI: 10.1007/s00209-018-2173-3.
► "Fano threefolds as equivariant compactifications of the vector group" (arXiv:1802.08090)
Joint with Zhizhong Huang at Beijing.
Michigan Mathematical Journal, 69 (2), pages 341-368, 2020.
DOI: 10.1307/mmj/1576033218.
► "A characterization of some Fano 4-folds through conic fibrations" (arXiv:1803.09129)
Joint with Eleonora Anna Romano at Genova.
International Mathematics Research Notices, 2021 (16), pages 12009-12036, 2021.
DOI: 10.1093/imrn/rnz244.
► "Equivariant compactifications of vector groups with high index" (arXiv:1806.02010)
Joint with Baohua Fu at Beijing.
Comptes Rendus Mathématique, 357 (5), pages 455–461, 2019.
DOI: 10.1016/j.crma.2019.05.002.
► "On the liftability of the automorphism group of smooth hypersurfaces of the projective space" (arXiv:2004.12455)
Joint with Víctor González-Aguilera at Santiago and with Álvaro Liendo at Talca.
Israel Journal of Mathematics, 255, pages 283-310, 2023.
DOI: 10.1007/s11856-022-2417-0.
► "Projective manifolds whose tangent bundle is Ulrich" (arXiv:2108.13944)
Joint with Vladimiro Benedetti at Nice, Yulieth Prieto at Trieste and Sergio Troncoso at Turin.
Journal of Algebra, Volume 630, 15 September 2023, Pages 248-273.
DOI: 10.1016/j.jalgebra.2023.03.046.
► "Del Pezzo quintics as equivariant compactifications of vector groups" (arXiv:2209.04152)
Joint with Adrien Dubouloz at Poitiers and Takashi Kishimoto at Saitama.
Algebraic Geometry, 11 (4), pages 593-619, 2024.
DOI: 10.14231/AG-2024-018.
► "On a Torelli Principle for automorphisms of Klein hypersurfaces" (arXiv:2212.13308)
Joint with Víctor González-Aguilera at Santiago, with Álvaro Liendo at Talca, and Roberto Villaflor at Valparaíso.
Transactions of the American Mathematical Society, Volume 377 (8), pages 5483-5511, 2024.
DOI: https://doi.org/10.1090/tran/9111.
► "On strictly elliptic K3 surfaces and del Pezzo surfaces" (arXiv:2312.14722)
Joint with Paola Comparin at Temuco, Yulieth Prieto at Trieste and Sergio Troncoso at Turin.
► "Constant Q-curvature metrics on a product Riemannian manifold" (arXiv:2402.14675)
Joint with Salomón Alarcón at Valparaíso, Simón Masnú at Santiago and Carolina Rey at Valparaíso.
► "Counting rational points on Hirzebruch-Kleinschmidt varieties over number fields" (arXiv:2407.19408)
Joint with Sebastián Herrero at Santiago and Tobías Martínez at San Miguel and Valparaíso.
► "Counting rational points on Hirzebruch-Kleinschmidt varieties over global function fields" (arXiv:2408.07631)
Joint with Sebastián Herrero at Santiago and Tobías Martínez at San Miguel and Valparaíso.
Currently, I am partially supported by the following grants:
► "Birational geometry and vector bundles on varieties with many automorphisms"
(Fondecyt Regular 1231214, Principal Investigator, 2023 - 2026).
Together with Álvaro Liendo as Co-Investigator.
► "On the automorphism group of quasi-smooth hypersurfaces of projective varieties with torus action"
(Fondecyt Regular 1240101, Co-Investigator, 2024 - 2027).
Together with Álvaro Liendo (PI) and Roberto Villaflor Loyola (CI).
► "Moduli, Windows and Cohomology"
(Fondecyt Postdoc 3240013, Sponsor, 2024 - 2026).
Together with Sebastián Torres (PI).
► "Equivariant Algebraic Geometry with a view towards Birational Geometry and Arithmetics"
(ANID-ECOS Project ECOS230044, Chilean Coordinator, 2024 - 2026).
Together with Álvaro Liendo and Giancarlo Lucchini-Arteche in Chile, and Adrien Dubouloz (French Coordinator), Enrica Floris and Ronan Terpereau in France.
Previously, I was supported by the following grants:
► "Linear systems and birational geometry of varieties of Fano type"
(Fondecyt Iniciación 11190323, Principal Investigator, 2019 - 2021).
► "Normal varieties with torus actions: equivariant compactifications of certain solvable groups and singularities"
(Fondecyt Regular 1200502, Co-Investigator, 2020 - 2023).
Together with Álvaro Liendo (PI) and Maximiliano Leyton-Álvarez (CI).
► "Topology of algebraic surfaces of general type and Punctual Quot scheme over surfaces"
(Fondecyt Postdoc 3210518, Sponsor, 2021 - 2023).
Together with Sergio Troncoso (PI).
Teaching
MAT529 ("Aritmética", i.e., Teoría Algebraica de Números): (2023-1).
MAT426 (Curvas Algebraicas): (2020-2), (2021-2), (2023-2). ¿Te interesa? Haz click aquí.
MAT290 (Geometría Diferencial): (2022-2).
MAT250 (Topología): (2024-1).
MAT235 (Variable Compleja): (2021-2), (2022-2).
MAT214 (Estructuras Algebraicas): (2019-1), (2021-1), (2023-1), (2024-1).
MAT210 (Álgebra Lineal): (2020-1).
MAT060 (Álgebra y Geometría): (2019-1), (2022-1).
Students
Graduate:
► Felipe Valladares (Master student, co-directed by Gabriele Ranieri).
Introducción a la Teoría de Esquemas Afines (15/01/2020).
► Fabián Levicán (Master student at PUCV, co-directed by Sebastián Herrero).
Potential density of rational points on elliptic K3 surfaces (07/12/2023).
► Tobías Martínez (PhD student, co-directed by Sebastián Herrero).
Counting rational points on Hirzebruch–Kleinschdmidt varieties over global fields (27/08/2024).
► Juan Fuenzalida (Master student at PUCV, co-directed by Sebastián Herrero).
Kodaira dimension of moduli spaces of polarized abelian varieties (05/07/2024).
► Sebastián Fuentes (Master student).
K-polistability of higher dimensional Fano varieties (ongoing).
Undergraduate:
► Eduardo Núñez.
Teorema de los números primos (10/12/2020).
► Gustavo Arcaya.
Einstein metrics and Geometry of the complex Monge-Ampère equation (09/06/2022).
► Fabián Levicán.
A computational approach to classification of additive smooth Fano polytopes (20/01/2023).
► Cristóbal Montecino.
Quantum cohomology, Gromov-Witten invariants and String theory (ongoing).
► Lucas Montero (co-directed by Dardo Goyeneche).
Algebraic geometry codes applied to Quantum information (ongoing).
► Emilio Oyanedel (co-directed by Roberto Villaflor).
Degree of irrationality and Vector bundles on K3 and Enriques surfaces (ongoing).
Participation as co-advisor (with another colleague as main advisor):
► Sofía Madrid (Undergraduate, directed by Francisco Cuevas).
Application of Topological Data Analysis to the distribution of a forest (ongoing).
► Mario Pastrana (Undergraduate, directed by Roberto Villaflor).
The Hodge conjecture for Klein hypersurfaces (ongoing).
► Cristian Pérez (Undergraduate, directed by Roberto Villaflor).
Punctual Torelli principle for non-extremal Klein hypersurfaces (ongoing).
Research assistant (i.e., Introduction to advanced topics in Algebraic Geometry):
► Gustavo Arcaya.
Hodge Theory on Kähler manifolds (2020).
► Cristóbal Montecino.
Enumerative geometry, Quantum cohomology and String theory (2021).
► Emilio Oyanedel.
K3 surfaces and Torelli Theorem (2022).
► Sebastián Fuentes.
Complex Algebraic Surfaces with a view towards the Minimal Model Program (2022).
► Mario Pastrana.
Complex Algebraic Surfaces with a view towards the Minimal Model Program (2022).
► Sebastián Fuentes.
Abelian Varieties and their Derived Categories (2023).
► Mateo Hidalgo.
Abelian Varieties and their Derived Categories (2023).
► Nolberto Rivera.
Moduli Problems and Geometric Invariant Theory (2024).
► Mateo Hidalgo.
Moduli Problems and Geometric Invariant Theory (2024).
► Cristian Pérez.
Hodge Theory for Projective Hypersurfaces (2024, ongoing).
► Barinia Muñoz.
Introduction to the Kähler-Ricci flow (2024, ongoing).
About me
A brief CV
Born on February 28th, 1991 in Copiapó (Chile)
Since March 2023: Assistant Professor, Universidad Técnica Federico Santa María, Valparaíso, Chile.
2018-2023 : Academic (Profesor Instructor), Universidad Técnica Federico Santa María, Valparaíso, Chile.
2018 : Postdoctoral Researcher at AMSS in the Chinese Academy of Sciences, Beijing.
2017 : ATER at Université Grenoble Alpes, Grenoble.
2014-2017 : Université Grenoble Alpes, Grenoble, PhD in Mathematics.
2013-2014 : Université Joseph Fourier, Grenoble, Master 2 Research in Mathematics.
2011-2013 : Universidad Técnica Federico Santa María, Valparaíso, Master in Mathematics.
2009-2011 : Universidad Técnica Federico Santa María, Valparaíso, Undergraduate degree in Mathematics.
2008 : Colegio Salesiano Valparaíso, Valparaíso, scientific formation.
Theses in mathematics
- PhD thesis: « Geometry of singular Fano varieties and projective bundles over curves » .
Advisors : Stéphane Druel and Catriona Maclean. Institut Fourier, Grenoble.
Thesis defence : october 2017. - Master 2 thesis: « Rationnally connected varieties and their behavoir in family » .
Advisor : Catriona Maclean. Institut Fourier, Grenoble.
Thesis defence : june 2014. - Master thesis: « On the automorphism group of complete intersections of type (2,3) ».
Advisor : Víctor González-Aguilera. Universidad Técnica Federico Santa María, Valparaíso.
Thesis defence : july 2013. - Undergraduate thesis: « Complex algebraic surfaces of type K3 ».
Advisor : Víctor González-Aguilera. Universidad Técnica Federico Santa María, Valparaíso.
Thesis defence : july 2011. - Groupe de travail 2016-2017 sur les Champs algébriques, based on several references, including the book of Neumann Algebraic Stacks and Moduli of Vector Bundles.
- Groupe de travail 2015-2016 sur les Surfaces algébriques complexes, based on the book of Beauville Surfaces algébriques complexes.
- Groupe de travail 2014-2015 sur les Déformations et les Espaces de Modules, based on the book of Hartshorne Deformation Theory.
- Algebraic Geometry, Lipschitz Geometry and Singularities, Pipa, Brazil. December 2023.
- 22nd Affine Algebraic Geometry Meeting, Saitama, Japan. March 2024.
My recent activities
Here is a list of some of my recent activities.
Working groups
I actively participate in the Algebraic Geometry Seminar organized by Giancarlo Urzúa (PUC Chile). For further information, click here.
On 2024, Roberto Villaflor (UTFSM) and I, we organized a reading group on Classical Hodge Theory. During the semester 2024-1 we will discuss the basics on the topology of algebraic varieties and consequences of the Hodge decomposition theorem and we followed the lecture notes of Hossein Movasati and Roberto Villaflor. For further information, click here.
On 2023, Mateo Hidalgo (UTFSM), Sebastián Fuentes (UTFSM) and I, we organized a reading group on abelian varieties. During the semester 2023-2 we will discuss the basics on abelian varieties and their derived categorias and we followed the lecture notes of Olivier Debarre. For further information, click here.
On 2022, Sergio Troncoso (UTFSM), Tobías Martínez (PUCV-UTFSM-UV) and I, we organized a reading group on riemannian geometry. During the semester 2022-1 we will discuss the basics on riemannian geometry and we followed the lecture notes of Olivier Biquard. For further information, click here.
On 2019, Gonzalo Manzano (USACH) and I, we organized a reading group on algebraic geometry. During the semester 2019-1 we discussed the basics on algebraic geometry and we followed the lecture notes of Joseph Le Potier. For further information, click here.
I organized, along with Louis-Clément, our own working group for PhD students. The goal was to read a book together and give a talk each week. Every year we worked on a quite large topic around complex algebraic geometry.