WIAG 2027 Projects

Description: Two kinds of object live on the moduli space M_{g,n}-bar of stable n-pointed curves of genus g, and both are controlled by the combinatorics of its boundary. 

The first is the cohomology of the space itself, together with the action of the symmetric group S_n permuting the marked points. The boundary of M_{g,n}-bar is stratified by stable graphs, and a great deal of cohomological information can be assembled by summing over graphs. Doing this correctly requires care with graph automorphisms, and there is now a systematic calculus for it.

The second is a family of vector bundles called conformal blocks. These sheaves depend on representations (of a Lie algebra or of a vertex operator algebra) attached to each marked point. One of the main properties of conformal blocks, called factorization, is again a statement about the boundary: when a curve degenerates, the fiber decomposes as a sum over ways of labelling the two branches at each node with dual modules. In good cases the Chern classes of these bundles are given by an explicit sum over stable graphs whose half-edges carry module labels.

So both objects are "graph sums," and the graphs in question are the same graphs. The project asks what happens when the two are combined: what is the S_n-equivariant content of the family of conformal blocks classes, and can it be computed by the same enumeration machinery that computes the cohomology of the ambient space? We will start from the smallest cases, where the labelled graphs and the resulting classes are both explicit enough to compute by hand.

Preferred background: Familiarity with moduli of curves and their boundary; basic representation theory of the symmetric group; no prior knowledge of conformal blocks or vertex operator algebras is required.

References:
[1] J. Kock and I. Vainsencher, An invitation to quantum cohomology, Progress in Mathematics, 249, Birkhäuser Boston, Boston, MA, 2007.
[2] R.P. Stanley, Enumeration Under Group Action in Algebraic combinatorics, Undergraduate Texts in Mathematics, Springer, Cham, 2018.
[3] A. Marian, D. Oprea, R. Pandharipande, A. Pixton and D. Zvonkine, The Chern character of the Verlinde bundle over $\overline{\mathcal{M}}_{g,n}$, J. Reine Angew. Math. 732 (2017), 147--163.
[4] C. Damiolini, A., C. Gibney and N. Tarasca, Vertex algebras of CohFT-type, in Facets of algebraic geometry. Vol. I, 164--189, London Math. Soc. Lecture Note Ser., 472, Cambridge Univ. Press, Cambridge.
[5] N. Fakhruddin, Chern classes of conformal blocks, in Compact moduli spaces and vector bundles, 145--176, Contemp. Math., 564, Amer. Math. Soc., Providence, RI.
[6] A. Gibney and S. Mukhopadhyay, On higher Chern classes of vector bundles of conformal blocks, arXiv:1609.04887.

Description: The aim of this project is to study and classify smooth genus g curves that arise as limits of smooth planar curves, built on work initiated in [1].  Classically, any smooth curve of genus 3 is either a plane quartic or hyperelliptic, and the existence of hyperelliptic limits may be explained by the existence of a mildly singular degeneration of P^2.  In [2] (or [1], using different methods), it is shown that the only possible smooth limits of plane quintics are either plane quintics or hyperelliptic curves, however in [1] is it shown that the only possible smooth limits of plane septics are plane septics.  We will study smooth limits of planar curves of higher degree, using the techniques in [1] which rely on moduli of pairs introduced in [3] and [4].

Depending on the interests of participants, connections to other areas may be studied (Brill-Noether loci, curves on other surfaces, higher dimensional analogues, etc). 

Background: Necessary background includes fundamentals on smooth curves at the level of Hartshorne chapter IV, and useful but not required background would be familiarity with gonality and existence of g^r_d's on curves, the geometry of M_g, KSBA moduli of pairs, and singularities of plane curves. 

References:
[1] K. DeVleming and D. Stapleton. Smooth limits of plane curves of prime degree and Markov numbers. J. Ec. polytech. Math., 11:683–731, 202.

[2] E. E. Griffin, Families of quintic surfaces and curves, Compositio Mathematica, 55 (1985), pp. 33–62.

[3] P. Hacking, Compact moduli of plane curves, Duke Math. J., 124 (2004), pp. 213–257.

[4] J. Kollár. Families of varieties of general type, volume 231 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2023. With the collaboration of K. Altmann and S. Kovács.

Description: There are many instances of Fano varieties with naturally associated Hyperkähler manifolds, e.g. cubic fourfold and Fano varieties of lines, Gushel-Mukai fourfolds and double EPW sextics, and Peskine sixfolds and Debarre-Voisin fourfolds. The geometry of the first is historically well-studied, while the latter is still a bit mysterious. As the Fano variety becomes more special in moduli, unexpected birationalities occur between the various Hyperkähler manifolds. This mirrors Hodge-theoretic associations between the varieties involved. We plan to exploit this connection to explore the geometry of one of these families (dependent on participant interests), using tools such as automorphisms, the derived category, and lattice theory.

Preferred background, any of the following: some background in derived categories of varieties, some background in Hyperkähler geometry, some background in Grassmannian/flag geometry, some background in advanced multi-linear algebra.

References:
[1] O. Debarre. Gushel-Mukai varieties, EMS Math. Surv. Sci., to appear (2026), available at arXiv:2001.03485.

[2] B. Hassett. Cubic fourfolds, K3 surfaces, and rationality questions. Rationality Problems in Algebraic Geometry (2016).

[3] S. Frei, C. Brooke, L. Flapan, and L. Marquand. Peskine sixfolds and Debarre-Voisin fourfolds with associated cubic fourfolds. Math. Nachr. (2026).

[4] S. Frei, C. Brooke, L. Marquand. Cubic fourfolds with birational Fano varieties of lines. Available at arXiv:2410.22259. 

[5] S. Billi, A. Grossi, and L. Marquand. Cubic fourfolds with a symplectic automorphism of prime order. Can. Math. Bull. (2026)

Description: The Chow group CH_0(X) of 0-cycles on a smooth projective variety X over a field k is a direct generalization to higher dimensions of the Picard group of a curve. It comes equipped with a two-step filtration CH_0(X) ≥ A_0(X) ≥ T(X) ≥ 0, where A_0(X)=CH_0(X)^{deg=0} is the subgroup of cycles of degree 0, and T(X) is the kernel of the Albanese map, alb_X: A_0(X) -> Alb_X(k), a higher dimensional analog of the Abel-Jacobi map for curves. A famous conjecture of S. Bloch predicts that for a smooth projective surface X with geometric genus p_g(X)=0 over an algebraically closed field k, the subgroup T(X) is trivial. It is classically known by [BKL76] that the conjecture is true for surfaces not of general type. In more recent years, it has been verified also for most known surfaces of general type with p_g(X)=0 (see for example [PW16]). 

Note that for a smooth projective surface X over an arbitrary field k, the vanishing of the Albanese kernel over the algebraic closure of k implies that T(X) is a torsion group. The primary purpose of this project will be to focus on certain classes of p_g=0 surfaces and obtain information on the structure of T(X) over non-algebraically closed fields; namely we will explore whether this group is torsion of finite exponent, and if yes we will try to give an upper bound on this exponent. On a different but related direction, another potential goal of this project will be to construct new examples of threefolds X for which we can verify the higher dimensional analog of Bloch's conjecture. This predicts that for a variety X of dimension d ≥ 2 over an algebraically closed field k, if dim_k H^0(X, Ω^p)=0 for all 2 ≤ p ≤ d, then T(X)=0. 

Suggested Background: This project will be most accessible to applicants who have some basic experience with algebraic surfaces. More important than specific background is the applicant's willingness to learn new material and work collaboratively as a team. 

References:
[BKL76] S. Bloch, A. Kas and D. Lieberman, Zero-cycles on surfaces with pg = 0. Compositio Math. 33, no. 2 (1976), 135-145.
[PW16] C. Pedrini and C. Weibel, Some surfaces of general type for which Bloch’s Conjecture holds. London Math. Soc. Lecture Note Ser., 427, Cambridge University Press, Cambridge, 2016. Arxiv: 1304.7523v1
[RS00] W. Raskind and M. Spiess, Milnor K-Groups and Zero-Cycles on Products of Curves over p-Adic Fields. Compositio Math. 121, 1-33 (2000).

Description: A principally polarized complex abelian variety is completely decomposable if it is isogenous to a product of elliptic curves. Ekedahl and Serre [4] asked in 1993 whether there is a curve whose Jacobian is completely decomposable in every genus g, and whether there is a bound on the genus for curves with completely decomposable Jacobians. Both questions remain open: the smallest genus with no known example is g=56 [5], and the largest dimension with a completely decomposable Jacobian is 1297 [4]. The standard machinery for producing such examples starts from a curve X with a group G of automorphisms and uses minimal left ideals of the rational group algebra Q[G] to split JX up to isogeny (the group algebra decomposition)  (see section 13.6 of [2]), and then apply other tools to the factors. See [1], [3], and [6] for a survey. 

Almost all of the explicit work in this area is done in terms of a period matrix (Z|D) with Z in the Siegel upper half space and D a diagonal matrix encoding the polarization type. This project proposes to redo the theory in the equivalent language of complex structures, roughly speaking, square matrices J such that J^2=-I, compatible with the polarization.

We expect this new approach to clarify certain aspects of the decomposition theory and yield results and examples that the period matrix strategy has not yet reached.

Prerequisites: A solid background in linear algebra and complex analysis is required. Basic aspects of the linear representation theory of finite groups are good to know, along with a working knowledge of Riemann surfaces and algebraic curves. Some prior exposure to the Jacobian variety, while not strictly necessary, will be helpful in following the more advanced topics discussed.

References

[1] R. Auffarth, H. Lange, and A. M. Rojas. A criterion for an abelian variety to be non-simple. Journal of Pure and Applied Algebra, 221(8):1906–1925, 2017.

[2] C. Birkenhake and H. Lange. Complex Abelian Varieties, volume 302 of Grundlehren der mathematischen Wissenschaften. Springer, 2nd edition, 2004.

[3] A. Carocca and R. E. Rodrı́guez. Jacobians with group actions and rational idempotents. Journal of Algebra, 306(1):322–343, 2006.

[4] T. Ekedahl and J.-P. Serre. Exemples de courbes algébriques à jacobienne complètement décomposable. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics, 317(5):509–513, 1993.

[5] J. Paulhus and A. V. Sutherland. Completely decomposable modular Jacobians. arXiv preprint arXiv:2502.16007, 2025.

[6] R. E. Rodrı́guez and A. M. Rojas. A fruitful interaction between algebra, geometry, and topology: Varieties through the lens of group actions. Notices of the American Mathematical Society, 71(6):715–723, 2024.

Description: Geometric Invariant Theory (GIT) is a procedure for constructing quotients in algebraic geometry, and is the motivating example for the theta-stability conditions used to study algebraic stacks. For us, the input of a GIT problem will be a representation V of a complex reductive group G, together with a character θ of G. These data define an open subset V_θ^ss (G) ⊆ V called the semi-stable locus. We define the GIT quotient to be the stack quotient [V_θ^ss (G)/G]. When G is a torus, this quotient is a toric stack (sometimes a toric variety), and just as many algebro-geometric constructions can be made explicit in toric geometry, these same constructions can often be made explicit for GIT quotients.

This project has an expository goal and a research goal.

(1) Write notes that develop the GIT construction outlined above, with the goal of giving a beginning algebraic geometry student a hands-on way to use tools in birational geometry and moduli theory, as well as an alternative approach to toric geometry. While the basic definitions can be found in [2], not even semi-thorough notes exist. Participants will be encouraged to take their favorite concept or construction from Hartshorne or elsewhere and understand it in the context of GIT.

(2) Explore an open question in nonabelian GIT: are there any examples of triples (V, G, θ) where G is not isogeneous to a product of general linear groups and G acts on V_θ^ss (G) with zero-dimensional stabilizers? The classification of GIT triples where G acts on V_θ^ss (G) with zero-dimensional stabilizers was begun in [3]. We will try to advance this classification by using the SageMath package developed in [1].

Prerequisites: Familiarity with algebraic stacks offers no particular advantage. Some familiarity with either toric geometry or SageMath is helpful, but not required; interest in both is essential. Enthusiasm for writing math and interest in improving your ability as a writer is crucial.

References:
[1] R. Hanson and J. Martinez-Garcia. The CompGIT package: a computational tool for Geometric Invariant Theory quotients. 20205. Preprint arXiv:2506.19431.
[2] A. King. Moduli of representations of finite dimensional algebras. The Quarterly Journal of Mathematics, Volume 45, Issue 4, December 1994, pp. 515–530.
[3] R. Kurama, R. Li, H. Talbott, and R. Webb. Weyl-invariant subspaces are (usually) not generic. 2025. Preprint arXiv:2510.03963.

Description: Our group will study problems related to the geometry of flag varieties via a combination of geometric and combinatorial methods. One possible direction concerns their derived categories of coherent sheaves and quantum cohomology rings. Quantum cohomology is a deformation of ordinary cohomology; its product structure is built from the Gromov Witten invariants, which count rational curves in a variety X subject to prescribed constraints. The big quantum cohomology BQH(X), which contains information about all n-point Gromov-Witten invariants, is usually very hard to compute; whereas the small quantum cohomology QH(X), a quotient of BQH(X), is explicitly understood when X is a flag variety (see survey [3] and the references within). 

Dubrovin's conjecture predicts that the existence of a full exceptional collection in the bounded derived category of coherent sheaves D^b(X) on a smooth projective variety X is equivalent to the generic semisimplicity of its big quantum cohomology BQH(X). A more refined conjecture is Conjecture 1.3 in [2] (see also Conjecture 3.5 in [1]), which says that more specific descriptions of D^b(X) are given by certain properties of the small quantum cohomology QH(X). One  possible goal is to verify this conjecture when X is a flag variety, using techniques such as combinatorial analysis on quantum Bruhat graphs or Gröbner degeneration.

Background: Familiarity with derived categories or Schubert calculus would be a plus, but not required. Depending on the specific interests of the group participants, other problems may be studied.

References:
[1] A. Kuznetsov. Semiorthogonal decompositions in families. In ICM—International Congress of Mathematicians. Vol. 2. Plenary lectures, pages 1154–1200. EMS Press, Berlin, 2023.
[2] A. Kuznetsov and M. Smirnov. Residual categories for (co)adjoint Grassmannians in classical types. Compos. Math., 157(6):1172–1206, 2021.
[3] H. Tamvakis. Quantum cohomology of homogeneous varieties: a survey. Oberwolfach Reports, 4(2):1212–1219, 2007. In the workshop report “Algebraic Groups”.