WIAG 2024 Projects
Table of Contents
Abelian Covers and Constructions on Algebraic Geometry
An abelian cover of complex projective varieties f: X →Y is a finite (flat) Galois morphism with abelian Galois group. Abelian covers have proven to be a very useful tool in showing the existence of algebraic varieties with special properties and as a testing ground for conjectures. Their structure is well understood ([A-P], [P]) in terms of the so-called building data, a collection of effective divisors and of line bundles on the target variety Y satisfying certain conditions. All the geometrical properties of the cover can be explicitly recovered from the building data. This has made it possible, with ingenious choices of the building data, to produce examples of varieties whose existence was an open question (such as surfaces of general type with canonical map of degree > 2 onto a canonically embedded surface), to exhibit pathologies of the moduli space of varieties of general type and to give an alternative, much simpler, description of known constructions (such as 2-dimensional ball quotients).
We want to push further the use of abelian covers to study open questions. An interesting such question is understanding the range of numerical invariants of varieties of general type that admit the action of a fixed finite abelian group. Another one is to explicitly produce boundary components in the compactified moduli space of surfaces of general type for unbounded values of the numerical invariants. However depending on the specific interests of the group participants other questions may be addressed.
Preferred background: basic knowledge of algebraic geometry at the level of Hartshorne’s textbook and some familiarity with surface theory.
Bibliography:
[A-P] V. Alexeev, R. Pardini, Non-normal abelian covers, Compos. Math., 148, n.4 (2012), 1051-1084.
[P] R.Pardini, Abelian covers of algebraic varieties, J. Reine Angew. Math., 417 (1991), 191-213.
Del Pezzo Surfaces over Finite Fields
A smooth two dimensional projective variety Xd defined over a field k is called a del Pezzo surface if its anticanonical divisor -ωX is ample. The degree d of a del Pezzo surface is the self-intersection number of its canonical class and is always between 1 and 9. From a moduli theoretic point of view, the most interesting del Pezzo surfaces are those of degree less than five. János Kollár proved that over an arbitrary field k, X4 is isomorphic to the intersection of two quadrics in ℙ4 , X3 is isomorphic to a cubic surface in ℙ3, X2 is isomorphic to a hypersurface of degree four in weighted projective space ℙ(2,1,1,1), and X1 is isomorphic to a hypersurface of degree six in weighted projective space ℙ(2,1,1,1) [4]. These surfaces are all birationally equivalent to the projective plane over algebraically closed fields, but this is often not true over finite fields. There has been a large interest in studying del Pezzo surfaces over finite fields in the last ten years. Some areas include but are not limited to studying the degree of unirationality, counting the rational points, and counting and studying the rational lines, see for example [1], [2], [3], [5], [6], [7], [8], [9], [10]. Some of the papers involving del Pezzo surfaces over finite fields avoid fields of characteristic 2 and 3. Our working group will seek to fill in some of the holes others have left open by avoiding the smallest characteristics.
Bibliography:
[1] B. Banwait, F. Fité, and D. Loughran, Del Pezzo surfaces over finite fields and their Frobenius traces, Math. Proc. Camb. Philos. Soc. 167 (1) (2019) 35–60.
[2] N. Kaplan, Rational point counts for del Pezzo surfaces over finite fields and coding theory, Harvard Ph.D. Thesis (2013).
[3] F. Karaoglu, and A. Betten, The number of cubic surfaces with 27 lines over a finite field. J. Algebraic Combin. 56 (2022), no. 1, 43–57.
[4] J. Kollár. Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 1996.
[5] A. Knecht, Degree of unirationality for del Pezzo surfaces over finite fields. J. ThˇZor. Nombres Bordeaux 27 (2015), no. 1, 171–182.
[6] A. Knecht, and K. Reyes, Full degree two del Pezzo surfaces over small finite fields, in: Contemporary Developments in Finite Fields and Applications (2016), pp. 145–159.
[7] C. Salgado, D. Testa, and A. V ́arilly-Alvarado, On the unirationality of del Pezzo surfaces of degree 2. J. Lond. Math. Soc. (2) 90 (2014), no. 1, 121–139.
[8] A. Trepalin, Del Pezzo surfaces over finite fields, Finite Fields Appl. 68 (2020) 32pp.
[9] A. Trepalin, Minimal del Pezzo surfaces of degree 2 over finite fields, Bull. Korean Math. Soc. 54 (5) (2017) 1779–1801.
[10] R. van Luijk, and R. Winter, Concurrent lines on del Pezzo surfaces of degree one. Math. Comp. 92 (2023), no. 339, 451–481.
Positivity of Toric Vector Bundles
Fujita's Conjecture states that if X is a smooth projective variety and L is an ample line bundle on X, then (1) "freeness": for k ≥ dim(X) + 1, Lk ⊗ ωX is globally generated, and (2) "ampleness": for k ≥ dim(X) + 2, Lk ⊗ ωX is very ample. Fujita's conjecture has been proven for various X; in particular, if X is a toric variety, it is true. Consider the case when the variety is the scroll ℙ(ℰ), with ℰ a toric vector bundle on a smooth toric variety X. One can then investigate ℙ(ℰ) and ask if Fujita's conjecture holds. For example, if ℰ is rank 2 on a toric variety, then ℙ(ℰ) satisfies the freeness part of the conjecture, due to work of Altmann and Ilten, [1]. Recent work of George and Manon have shown Fujita's conjecture holds for several classes of projectivized toric bundles [2], in particular for the projectivization ℙ(ℰ) of any rank n irreducible toric vector bundle on ℙn, [3].
In this project we will investigate projectivized toric vector bundles for other classes of toric vector bundles, with the hope of verifying Fujita's conjecture. For a toric vector bundle ℰ on a toric variety X, sections of line bundles on ℙ(ℰ) can be described combinatorially using the parliament of polytopes associated to ℰ. More precisely, Klyachko gives a description of toric vector bundles by associating to each ray in the fan, a decreasing filtration of vector spaces, [5], and using his description, di Rocco, Jabbusch and Smith define a matroid which gives a parliament of polytopes, that is a collection of polytopes associated to a given vector bundle, [4]. This generalizes the notion of a polytope associated to a line bundle on a toric variety. Moreover, the parliament of polytopes encodes positivity of the toric vector bundle, and from the parliament of polytopes, one can read off the global sections of a toric vector bundle. The hope is that we can use the parliament of polytopes to study sections of line bundles on ℙ(ℰ) for various toric vector bundles ℰ.
Bibliography:
[1] Klaus Altmann and Nathan Ilten, Fujita’s freeness conjecture for T-varieties of complexity one, Michigan Math. J. 69 (2020), no. 2, 323–340, DOI 10.1307/mmj/1574326879.
[2] Courtney George and Christopher Manon, Positivity properties of divisors on Toric Vector Bundles, 2023. Preprint arXiv:2308.09014.
[3] Courtney George and Christopher Manon, Algebra and Geometry of Irreducible toric vector bundles of rank n on ℙn, 2023. Preprint arXiv:2308.09017.
[4] Sandra Di Rocco, Kelly Jabbusch, and Gregory A. Smith, Toric vector bundles and parliaments of polytopes, Trans. Amer. Math. Soc. 370 (2018), no. 11, 7715–7741, DOI 10.1090/tran/7201.
[5] A.A. Klyachko, Equivariant bundles over toric varieties, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 5, 1001–1039, 1135 (Russian); English transl., Math. USSR-Izv. 35 (1990), no. 2, 337–375.
Chow rings of Hurwitz spaces with marked ramification
Let Hk,g be the Hurwitz space parametrizing simply branched degree k covers C → ℙ1 where C is a smooth curve of genus g. The theory of admissible covers [4] gives rise to a compactification H̅k,g ⊃ Hk,g. The moduli space H̅k,g has a natural stratification by the topological type of the source curve. The largest stratum is Hk,g, the covers of smooth curves. Meanwhile, the boundary strata of Hk,g ⊂ H̅k,g are built by gluing together smooth covers along fibers with matching ramification profiles. This motivates Hurwitz spaces with marked ramification Hk,g(μ1,...,μn), which parameterize covers together with marked fibers having ramification profiles μi. The locally closed boundary strata of Hk,g ⊂ H̅k,g are all finite group quotients of products of various Hk,g(μ1,...,μn). For k = 2, the rational (and in some cases integral) Chow rings of Hk,g(μ1,...,μn) are known, provided n is small relative to g [1,3]. For k = 3, only the case n = 0 — that is Hk,g itself — is understood [2]. The goal of this project is to study the Chow rings of these moduli spaces when k = 3 and n > 0. As an application, we hope to prove results about the low-codimension Chow groups of H̅3,g.
Bibliography:
[1] S. Canning and H. Larson, The rational Chow rings of moduli spaces of hyperelliptic curves with marked points, to appear in Int. Math. Res. Not., https://arxiv.org/pdf/2207.10873.pdf.
[2] S. Canning and H. Larson, Chow rings of low-degree Hurwitz spaces, Crelle, vol. 2022, no. 789 (2022) pp. 103-152, https://arxiv.org/abs/2110.01059.
[3] D. Edidin and X. Hu, The integral Chow rings of the stacks of hyperelliptic Weierstrass points, https://arxiv.org/pdf/2208.00556.pdf.
[4] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23–88
Experiments with Ceresa cycles of low genus curves
The Ceresa cycle is a canonical algebraic cycle associated to an algebraic curve. It is known to vanish for hyperelliptic curves, and until recently it was unknown whether there are any nonhyperelliptic curves with torsion Ceresa cycle. Over the last few years, there have been many new examples of non-hyperelliptic curves with torsion Ceresa cycle ([BLLS22, QZ22, LS23]). In this project, we will explore sufficient arithmetic and geometric conditions that guarantee the triviality/nontriviality of the Ceresa cycle, in families of low genus curves equipped with additional geometric structure, including some examples with curves over finite fields.
Preferred background: This project will be most accessible to applicants who have some experience with algebraic curves over number fields and/or finite fields. Topics that will come up from algebraic geometry include divisors of functions. Topics that will come up from number theory include the Hasse-Weil bound. We may run basic experiments on computer algebra systems such as SageMath or Magma. More important than specific background is the applicant's willingness to learn new material and work collaboratively as a team.
Bibliography:
BLLS22 -- Bisogno-Li-Litt-Srinivasan, Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class
LS23 -- Lillienfeldt-Shnidman, Experiments with Ceresa classes of cyclic Fermat quotients
QZ22 -- Qiu-Zhang, Vanishing results in Chow groups for the modified diagonal cycles
Geometry and topology of Springer fibers and Hessenberg varieties
The Springer fiber of a matrix X is the subvariety of the flag variety consisting of flags fixed by X. Springer fibers are the fibers of a particular smooth resolution of the nilcone, namely the subspace of nilpotent matrices inside the collection of n x n matrices. They are named after Tonny Springer, who discovered a beautiful representation of the symmetric group on the cohomology of Springer fibers and showed that when X is nilpotent, the top-dimensional cohomology is in fact the irreducible representation corresponding to the partition of n given by the sizes of the Jordan blocks of X. Hessenberg varieties generalize Springer fibers with an additional parameter h that regulates how much the matrix X can move each flag. Like Springer fibers, Hessenberg varieties are connected to deep questions in combinatorial and geometric representation theory.
The geometry of both Springer fibers and Hessenberg varieties is connected to the combinatorics of permutations (from the flag variety) and partitions (from the Jordan form of X). Their geometric and topological structure is rich in its own right, and also informs questions in representation theory, knot theory, and combinatorics. Yet relatively little is known about it, especially compared to Schubert varieties, which are similar subvarieties of the flag variety. This project will attack open questions about the geometry and topology of Hessenberg varieties.