Pablo Magni

Articles

• Kernels of categorical resolutions of nodal singularities. Joint with Warren Cattani, Franco Giovenzana, Shengxuan Liu, Luigi Martinelli, Laura Pertusi, and Jieao Song. Rend. Circ. Mat. Palermo, II. Ser (2023).

  In this paper we study derived categories of nodal singularities. We show that for all nodal singularities there is a categorical resolution whose kernel is generated by a 2 or 3-spherical object, depending on the dimension. We apply this result to the case of nodal cubic fourfolds, where we describe the kernel generator of the categorical resolution as an object in the bounded derived category of the associated degree six K3 surface.

• Derived equivalences of generalized Kummer varieties. Preprint, 2022.

  In this article we study derived (auto)equivalences of generalized Kummer varieties Kumⁿ(A). We provide an answer to a question raised by Namikawa by showing that the generalized Kummer varieties Kumⁿ(A) and Kumⁿ(A^∨) are derived equivalent as long as n is even and the abelian surface A admits a polarization whose exponent is coprime to n+1. Furthermore we obtain exact sequences involving groups of autoequivalences in the style of Orlov's short exact sequence for autoequivalences of abelian varieties.

Thesis:

• Derived equivalences of generalized Kummer varieties. PhD thesis, supervised by Lenny Taelman and Lie Fu, defended in September 2023.
• Finiteness results and the Tate conjecture for K3 surfaces via cubic fourfolds. Master's thesis, supervised by Daniel Huybrechts, 2018.

  We explore the arithmetic of Hassett’s association, which assigns K3 surfaces to special cubic fourfolds. We demonstrate that this association, which a priori is transcendental, descends to number fields.

  The motivation for this is the Tate Conjecture for K3 surfaces over finite fields, and its relation to finiteness statements. More precisely, by a Theorem of Lieblich–Maulik–Snowden, the Tate Conjecture for K3 surfaces over finite fields is true if and only if for each fixed finite field there exist only finitely many K3 surfaces over this field. Now, the idea is that we embed the moduli spaces of polarized K3 surfaces into a fixed moduli space (in our case the moduli space of cubic fourfolds) which has only finitely many points over a finite field.

  We work towards a proof of the Tate Conjecture in special cases using this approach via a finiteness statement. For this we continue Charles study of the proof of Lieblich–Maulik–Snowden’s theorem and discuss which (weaker) finiteness statements are sufficient to deduce the Tate Conjecture for a given K3 surface.

• p-adic integration and birational Calabi-Yau varieties. Bachelor's thesis, supervised by Daniel Huybrechts, 2016.

  This bachelor thesis is an exposition of Batyrev's theorem that two birationally equivalent Calabi-Yau varieties have equal Betti numbers.

  The proof is based on methods of p-adic analysis, especially p-adic integration on the analytification of smooth varieties over a p-adic field. By a theorem of Weil one can relate the volume of these p-adic analytic manifolds with the number of points in a reduction to a finite field of the variety under consideration. This reduces the comparison of local zeta functions (and in particular Betti numbers, by the Weil conjectures) to the comparison of volumes of certain p-adic analytic manifolds. Batyrev’s theorem follows by showning that the p-adic volumes of two birationally equivalent Calabi–Yau varieties coincide.

  In order to realize the indicated strategy formally, we work out a few technicalities. In particular, the method of “spreading out” is explained, which lets us move from varieteis over the complex numbers to varieties over a p-adic field.

Talks, Seminars, Conferences

Invited talks:

• Derived equivalences of generalized Kummer varieties. Fano varieties and Hyper-Kähler varieties. Strasbourg, January 2023
• Derived equivalences of generalized Kummer varieties. AMS-SMF-EMS Joint International Meeting: Special Session on Derived Categories and Rationality. Grenoble, July 2022

Conferences attended:

• Integral Structures in Geometry and Representation Theory, Paderborn, September 2024
• CRC Retreat 2024, Bad Sassendorf, February 2024
• HIM Junior Trimester Program: Derived categories, Hodge Theory, and Chow Groups, Bonn, September-December 2023
• JTP Birational Workshop, Bonn, November 2023
• JTP Derived Workshop, Bonn, November 2023
• Categories and Stacks in Algebraic Geometry and Algebraic Topology, CIRM, October 2023
• Panorama of Mathematics II, Bonn, October 2023
• School: K3 surfaces, hyperkähler manifolds, and cubic fourfolds, Bonn, September 2023
• Fano varieties and Hyper-Kähler varieties, Strasbourg, January 2023
• HyperKähler Manifolds and Related Geometries, Cetraro, August 2022
• AMS-SMF-EMS Joint International Meeting: Special Session on Derived Categories and Rationality, Grenoble, July 2022
• New Perspectives on Hyperkähler Manifolds, Levico Terme, June 2022
• 43rd Autumn School in Algebraic Geometry: K3 categories and hyperkähler moduli spaces, Lukecin (Online), September 2021
• Hausdorff School: “Hyperkähler Geometry”, Bonn, September 2021
• CIMPA Research School: Hodge Theory and p-adic Hodge Theory, CIMAT (Online), August 2021
• GQT Graduate School, Online, January 2021
• Masterclass in Condensed Mathematics, Copenhagen (Online), November 2020
• IHES Summer School: Motivic, Equivariant and Non-commutative Homotopy Theory, Bures-sur-Yvette (Online), June 2020
• Belgian-Dutch Algebraic Geometry Day, Leuven, February 2020
• Workshop on Algebraic Geometry and Homotopy Theory, Nijmegen, October 2019
• GQT Graduate School and Colloquium, Den Dolder, July 2019
• Final Conference of the SFB/TRR45 Bonn/Essen/Mainz: Geometry and Arithmetic of Algebraic Varieties, Bonn, June 2019
• Summer School: Motives Today, Trento, June 2019
• Kleine AG: Serre's Modularity Conjecture, Bonn, May 2019
• Rationality of Algebraic Varieties, Schiermonnikoog Island, April 2019
• Kleine AG: Shimura Varietäten, Bonn, October 2018
• Conference on Arithmetic Algebraic Geometry: On the occasion of Michael Rapoport’s 70th birthday, Bonn, October 2018
• Belgian-Dutch Algebraic Geometry Seminar, Nijmegen, September 2018
• K3 surfaces and Galois representations, Shepperton, May 2018
• Kleine AG: Deformationen abelscher Varietäten, Bonn, April 2018
• School on Birational Geometry of Hypersurfaces, Gargnano del Garda, March 2018
• Kleine AG: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Heidelberg, October 2017
• Autumn school: Topics in arithmetic and algebraic geometry, Mainz, October 2017
• Students' Conference on Nonarchimedean and Tropical Geometry, Regensburg, August 2017
• Arbeitstagung 2017 on “Physical Mathematics”, Bonn, June 2017
• Bay Area Haskell Hackathon, San Francisco, April 2017
• Panorama of Mathematics, Bonn, October 2015.

Teaching

• Algebra I, Bielefeld WS24 (teaching assistant)
• Algebraic Geometry III, Bielefeld SS24 (teaching assistant)
• Geometrie (Gym/Ge), Bielefeld SS24 (teaching assistant)
• Riemann Surfaces, Mastermath course, Nijmegen SS21 (teaching assistant)
• Category Theory and Homological Algebra, Nijmegen WS20 (teaching assistant)
• Riemann Surfaces, Mastermath course, Nijmegen SS20 (teaching assistant)
• Galois Theory, Nijmegen WS19 (teaching assistant)
• Riemann Surfaces, Mastermath course, Nijmegen SS19 (teaching assistant)
• Lineare Algebra für Informatiker und Lehramt Mathematik, Bonn SS17 (teaching assistant)
• Analysis 2, Bonn SS15 (teaching assistant)