Algebraic Geometry (VI ELGA)

Speaker : Jarod ALPER (University of Washington,United States)

This lecture series explores recent advances in the construction and study of moduli spaces, moving beyond classical Geometric Invariant Theory (GIT) to the intrinsic geometry of algebraic stacks. Drawing from Chapters 7–9 of "Stacks and Moduli" by Jarod Alper, the series develops the theory of good moduli spaces as a robust alternative to GIT, particularly for objects with infinite automorphism groups. We will cover the geometry of algebraic stacks, the valuative criteria for $\Theta$- and S-completeness, and the existence theorem for good moduli spaces. Finally, we will apply these tools to the moduli stack of semistable vector bundles on a curve, proving its projectivity through both intrinsic stack-theoretic methods and classical GIT. Bibliography: 1. Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry 2. Jarod Alper, Evolution of Stacks and Moduli 3. Jarod Alper, Stacks and Moduli 4. Deligne and Mumford, The irreducibility of the space of curves of given genus Details about the course development and topics will be announced soon. Therefore, the title and description of this are tentative for now.

Speaker : José GONZALEZ (University of California,Riverside,United States)

The Cox ring of a normal projective variety $X$ is a multigraded algebra that connects the birational geometry of $X$ to commutative algebra and geometric invariant theory. The Cox ring contains the section rings of all divisor classes on $X$ as subalgebras, and it generalizes the homogeneous coordinate rings of toric varieties. When $X$ is normal, projective, $\mathbb{Q}$-factorial, and with a finitely generated divisor class group, we say that $X$ is a Mori dream space (MDS) if it's Cox ring is finitely generated. The motivation is that in this case $X$ satisfies many desirable properties in the minimal model program. For example, on an MDS the cones of pseudoeffective, nef, and movable divisors of $X$ are rational polyhedral, and one can run a minimal model program for every effective divisor. Moreover, the movable cone admits a polyhedral chamber decomposition, and the resulting wall crossing controls the small $\mathbb{Q}$-factorial modifications of $X$. This course will introduce Cox rings and present their properties, examples, and applications. For instance, under the above assumptions on the varieties involved, the finite generation of the Cox ring is preserved under small $\mathbb{Q}$-factorial modifications, surjective morphisms, and good GIT quotients. On the other hand, the behavior of the MDS property under blowups is very subtle, but there are many interesting results. We will also see classes of examples that play a central role in current research, including blowups of toric or rational varieties and moduli spaces such as $\overline{M}_{0,n}$ and Fulton-MacPherson type compactifications, highlighting both finite generation results and mechanisms that lead to non-finitely generated Cox rings. Bibliography: 1. Finite Generation of Cox Rings (by Jose Gonzalez and Antonio Laface) 2. The first chapter of the book Cox rings (by Ivan Arzhantsev, Ulrich Derenthal, Jürgen Hausen, and Antonio Laface) 3. Mori dream spaces and GIT (by Yi Hu and Sean Keel) 4. The Cox ring of an algebraic variety with torus action (by Jurgen Hausen and Hendrik Suss) 5. Mbar_{0,n} is not a Mori dream space (by Ana-Maria Castravet and Jenia Tevelev) 6. The Fulton-MacPherson compactification is not a Mori dream space (Patricio Gallardo, Jose Gonzalez and Evangelos Routis

Speaker : Laura ESCOBAR (University of California,Santa Cruz,United States)

Modern Schubert calculus studies Schubert varieties and their cohomology classes via Schubert polynomials. Matrix Schubert varieties provide an affine computational framework, with multidegrees given by Schubert polynomials. Alternating sign matrices naturally index intersections of matrix Schubert varieties, called ASM varieties. This course develops the geometric theory of ASM varieties, culminating in recent work showing how weak order on ASMs provides tools for computing fundamental invariants like codimension and Castelnuovo-Mumford regularity. Bibliography: 1. W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J. 65 (1992), no. 3, 381–420. [For Lecture 1] 2. A. Knutson, E. Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), no. 3, 1245–1318. [For Lectures 1, 3] 3. Z. Hamaker, V. Reiner, Poset structures on alternating sign matrices, arXiv:2111.06482. [For Lecture 2] 4. L. Escobar, P. Klein, A. Weigandt, Algebra and geometry of ASM weak order, arXiv:2502.19266. [For Lecture 4]

Speaker : Lena JI (University of Illinois,Urbana-Champaign,United States)

A variety is said to be rational if it is birationally equivalent to projective space. In this course, we study at rationality of varieties in dimensions at most three. We will start in dimensions one and two over algebraically closed fields. Then we will move to non-closed fields and see how the arithmetic of the field comes into play. Finally, we will consider dimension three. We will explain some rationality obstructions for complex threefolds, and then discuss some recent generalizations of these obstructions to non-closed fields. Bibliography • Arnaud Beauville. The Lüroth problem. In Rationality problems in algebraic geometry, volume 2172 of Lecture Notes in Math., pages 1–27. Springer, Cham, 2016 • Olivier Debarre. On rationality problems. EMS Surv. Math. Sci., 2024. published online first. • Brendan Hassett. Rational surfaces over nonclosed fields. In Arithmetic geometry, volume 8 of Clay Math. Proc., pages 155–209. Amer. Math. Soc., Providence, RI, 2009. • Anthony Várilly-Alvarado. Arithmetic of del Pezzo surfaces. In Birational geometry, rational curves, and arithmetic, Simons Symp., pages 293–319. Springer, Cham, 2013.

Speaker : Cecilia SALGADO (University of Groningen,Netherlands)

In the study of abundance of rational points on algebraic varieties, two main tools play a prominent role, namely automorphisms and fibrations. This minicourse is dedicated to the latter. More precisely, to elliptic fibrations on surfaces. These have played a key part in the proof of results such as unirationality over arbitrary fields and Zariski density of rational points, over the past 25 years. In this mini-course of 4 lectures, we will treat geometric and arithmetic aspects of elliptic fibrations on algebraic surfaces. The first lecture will be dedicated to bringing everyone on board, covering the necessary background on algebraic curves, and some interesting examples of surfaces. Lectures 2 and 3 will discuss a motivation for this course arising from the study of rational points on algebraic varieties. They will also introduce elliptic surfaces, the Kodaira-Néron classification of singular fibers, the Shioda-Tate formula, the Néron-Tate height, the Mordell-Weil group of sections, torsion sections and Silverman's specialization theorem. The fourth lecture will be dedicated to studying elliptic fibrations as a tool to treat arithmetic questions such as unirationality and Zariski density as used in works of Bogomolov-Tschinkel, Hassett, Kollar-Mella and van Luijk-Salgado. References: Background on algebraic curves: 0. A. Gathmann, Plane algebraic curves, lecture notes available online 1. W.Fulton, Algebraic curves: An Introduction to Algebraic Geometry, Addison-Wesley (2008) available online 2. J.Silverman and J. Tate, Rational points on elliptic curves, Springer (2015) 3. J. Silverman, Arithmetic of elliptic curves, Springer (2009) Elliptic surfaces: 4. R. Miranda, The basic theory of elliptic surfaces, ETS editrice (1989) available online. 5. M. Schuett and T. Shioda, Elliptic surfaces, Adv. Stud. Pure Math., 2010: 51-160 (2010) Rational points via elliptic surfaces: 6. B.Hassett, Potential density of rational points on algebraic varieties, Higher Dimensional Varieties and Rational Points. Bolyai Society Mathematical Studies, vol 12. Springer (2003) 7. F. Bogomolov and Y. Tschinkel, On the density of rational points on elliptic fibrations. J. reine angew. Math., 511 (1999), 87–93. 8. J. Kollár, and M. Mella, Quadratic families of elliptic curves and unirationality of degree 1 conic bundles, American Journal of Math. 139 (2017) 9. C. Salgado, R. van Luijk, Density of rational points on del Pezzo surfaces of degree one, Advances in Mathematics (2014)

Speaker : Roberto SVALDI (Universita degli Studi di Milano,Italy)

This course aims to introduce the birational study of foliations on algebraic varieties. Over the past 25 years, the field has undergone substantial development, beginning with McQuillan’s program for classification of foliations on surfaces, inspired by Bogomolov’s approach to the Green–Griffiths–Lang conjecture in dimension two. More recent advances have sought to systematically compare the birational classification of foliations with the classical birational classification of varieties via the Minimal Model Program, revealing both striking analogies and fundamental differences. The course will survey the current state of the art, emphasizing how these similarities and divergences inform our understanding of foliations and how their interaction with birational geometry has shaped recent progress in the area. Bibliography: 1. M. Brunella, Birational geometry of foliations, IMPA Monogr., 1, Springer, Cham, 2015, xiv+130 pp. [For Lecture 1-2] 2. J. Pereira, R. Svaldi, Effective algebraic integration in bounded genus, Algebr. Geom. 6 (2019), no. 4, 454–485. [For Lecture 2-3] 3. C. Araujo, S. Druel, On Fano foliations, Adv. Math. 238 (2013), 70-118. [For Lecture 4] 4. F. Ambro, P. Cascini, V. Shokurov, C. Spicer, Positivity of the Moduli Part, arXiv e-print, arXiv:2111.00423v2 (and future versions). [For Lecture 3]