This is the page for the CRAAG Seminar at UGA. We’ll be reading classic papers in arithmetic and algebraic geometry (where the phrases “arithmetic geometry,” “algebraic geometry,” and “classic” are all broadly construed), presenting them, and discussing them.
Schedule 2021
6/4 — Organizational meeting
6/18 — Cancelled due to internet issues
6/25 — Nolan Schock, Tropical curves, graph complexes, and top weight cohomology of \(M_g\)
7/2 — Freddy Saia, Isogeny classes of abelian varieties over finite fields
7/9 — Zack Garza, On the de Rham cohomology of algebraic varieties
7/16 — Tyler Genao, Entanglement in the family of division fields of elliptic curves with complex multiplication
7/23 — Santana Afton, on Golod-Shafarevich inequalities and applications
7/30 — Borys Kadets, mystery topic
8/6 — Daniel Litt, Deligne on regular singular connections
8/13 — Arvind Suresh, Il n'y a pas de variété abélienne sur \(\mathbb{Z}\)
Schedule 2020
1/16 — Valery Alexeev on Projective manifolds with ample tangent bundles by Mori
1/23 — Jimmy Dillies on Un théorème de finitude pour la monodromie by Deligne
1/30 — Sasha Shmakov on Formal moduli for one-parameter formal groups by Lubin and Tate
2/6 — Ben Tighe on Relèvements modulo \(p^2\) et décomposition du complexe de de Rham by Deligne and Illusie
2/13 — Nikon Kurnosov on Specialization of birational types by Kontsevich and Tschinkel
2/20 — Changho Han on irreducibility of \(M_g\)
2/27 — Nolan Schock on Grothendieck-Riemann-Roch
3/5 — Phil Engel on The intermediate Jacobian of a cubic threefold
3/12 — Spring break
3/19 — Daniel Litt on A finiteness theorem for Galois representations of function fields over finite fields (after Deligne) (video available here)
3/26 — Nicholas George Triantafillou on Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin
4/2 — Arvind Suresh on Grothendieck’s Groupe de Brauer I
4/9 — Sasha Shmakov on Tate’s thesis (pre-talk slides, talk slides)
4/16 — Daniel Litt on Groupe de Brauer I (continuing from Arvind Suresh’s talk on 4/2)
4/23 — Zack Garza on Weil, Numbers of solutions of equations in finite fields, (slides)
List of papers
A preliminary list of papers can be found below, but feel free to suggest any paper you’re interested in, or to ask me for other suggestions. Please claim a paper and date — as you read the paper, you should come talk to me about how to best present it.
Araujo-Kollár, Rational curves on varieties — studies rational curves on algebraic varieties, building on “bend-and-break”
Atiyah, Vector bundles over an elliptic curve — a classification of vector bundles on an elliptic curve. A beautiful, fairly elementary paper, and a good prequel to the notion of stability for vector bundles.
Beauville-Voisin, On the Chow ring of a K3 surface — constructs a very interesting 0-cycle on K3 surfaces
Behrend-Fantechi, The intrinsic normal cone — Explains how to construct virtual fundamental classes in many settings
Buhler-Reichstein, On the essential dimension of a finite group — Introduces the notion of “essential dimension” and computes it in some cases.
Borel-Serre, Le théorème de Riemann–Roch, d'après A. Grothendieck — a proof of the Grothendieck-Riemann-Roch theorem. We should possibly follow this talk with one about an application of GRR. See translation here, by Tim Hosgood.
Chan-Galatius-Payne, Tropical curves, graph complexes, and top weight cohomology of \(M_g\) — uses tropical techniques to show that \(M_g\) has lots of non-tautological cohomology
Clemens-Griffiths, The intermediate Jacobian of a cubic threefold — shows that smooth cubic threefolds aren’t rational.
Ceresa, \(C\) is not algebraically equivalent to \(-C\) in its Jacobian — shows that for a generic curve of genus at least 3, the image of the curve under the Abel-Jacobi map is not algebraically equivalent to its inverse. This is interesting because these two cycles are homologically equivalent.
Deligne, Un théorème de finitude pour la monodromie — a finiteness result about representations of fundamental groups which underly (polarizable) variations of Hodge structure. This is the “function field” version of the Shafarevich conjecture.
Deligne-Illusie, Relèvements modulo \(p^2\) et décomposition du complexe de de Rham — a proof that the Hodge-to-de Rham spectral sequence degenerates in characteristic zero, via characteristic p methods; also, a purely algebraic proof of the Kodaira vanishing theorem.
Deligne-Mumford, The irreducibility of the space of curves of a given genus — proves that the moduli space of curves \(M_g\) is irreducible
Deninger-Werner, Vector bundles on \(p\)-adic curves and parallel transport — the beginning of non-abelian \(p\)-adic Hodge theory
Esnault-Kerz, A finiteness theorem for Galois representations of function fields over finite fields (after Deligne) — an l-adic variant of “Un théorème de finitude pour la monodromie.” A talk on this should focus on the case of curves (which is only sketched in the paper — come talk to me if you’re interested) and then spend a bit of time on how to deduce the case of higher-dimensional varieties.
Esnault-Mehta, Simply connected projective manifolds in characteristic \(p>0\) have no nontrivial stratified bundles — proves that two natural notions of “fundamental group” are closely related
Fontaine, Il n'y a pas de variété abélienne sur \(\mathbb{Z}\) — proves what it says in the title, namely that there are no Abelian schemes over \(\text{Spec}(\mathbb{Z})\).
Fontaine, Exposé II: Les corps des périodes \(p\)-adiques — develops the period rings arising in \(p\)-adic Hodge theory
Grothendieck, Hodge's general conjecture is false for trivial reasons — reformulates the generalized Hodge conjecture to avoid certain clever but “trivial” counterexamples.
Grothendieck, Le groupe de Brauer I-III — develops the theory of the Brauer group of a scheme. Incredibly fun and incredibly useful.
Grothendieck, On the de Rham cohomology of algebraic varieties — compares algebraic de Rham cohomology to the usual notion for manifolds, using resolution of singularities.
Hassett-Pirutka-Tschinkel, Stable rationality of quadric surface bundles over surfaces — shows that non-rationality does not specialize
Harer-Zagier, The Euler characteristic of the moduli space of curves — computes the Euler characteristic of \(M_g\)
Honda, Isogeny classes of abelian varieties over finite fields — classifies Abelian varieties over finite fields, up to isogeny, in terms of \(q\)-Weil numbers.
Katz, Algebraic solutions of differential equations (p-curvature and the Hodge filtration) — formulates the \(p\)-curvature conjecture and proves it in the case of the Gauss-Manin connection.
Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin — studies the local monodromy of the Gauss-Manin connection via mixed-characteristic methods. A great place to learn about algebraic differential equations.
Kontsevich-Tschinkel, Specialization of birational types — shows that rationality specializes (for families of smooth varieties), via a simple and beautiful argument (motivated by the notion of “motivic nearby cycles,” which is much more elementary than it sounds).
Lubin-Tate, Formal moduli for one-parameter formal groups — constructs special formal groups which realize “explicit class field theory” for local fields. A talk on this should give context from class field theory, e.g. from Cassels-Frohlich.
Mori, Projective manifolds with ample tangent bundles — proves Hartshorne’s conjecture, characterizing projective space as the only smooth projective variety with ample tangent bundle. Introduces “bend-and-break.”
Mumford, Picard groups of moduli problems — computes the Picard group of \(\mathscr{M}_{1,1}\), without exactly saying that this is what it’s doing. A great prequel to the notion of a stack.
Narasimhan-Seshadri, Stable and unitary vector bundles on a compact Riemann surface — the beginning of non-abelian Hodge theory
Tate, Fourier analysis in number fields, and Hecke's zeta-functions (Tate’s thesis) — Introduces the adelic point of view on \(L\)-functions, and proves functional equations/analytic continuation in many cases.
Tate, p-divisible groups — the beginning of the theory of \(p\)-divisible groups. A good prequel to \(p\)-adic Hodge theory.
Serre-Tate, Good reduction of abelian varieties — proves the Nèron-Ogg-Shafarevich criterion for good reduction of abelian varieties, using Nèron models.
Vakil, Murphy’s law in algebraic geometry — shows that moduli spaces of nice objects can be arbitrarily singular.
Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus — Proves Green’s conjecture in many cases
Voisin, On the homotopy types of compact Kähler and complex projective manifolds — provides the first examples of compact Kähler manifolds which are not homotopy equivalent to any complex projective variety.
Weil, Numbers of solutions of equations in finite fields — states the Weil conjectures and proves them for certain special hypersurfaces.