Pedro Montero

Since 2024, I am the Vice-President of the Chilean Mathematical Society (SOMACHI).

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.

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).

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).

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).