Research Statement
The general area of my research is algebraic geometry, more specifically I study stacky curves and moduli spaces of sheaves on them.
A stacky curve is a tame one-dimensional Deligne-Mumford stack over a field. All stacky curves can be glued together from finite quotients of curves so they can be thought of as orbifolds. They can also be constructed as root stacks, which allows us to think of them as a curve together with a set of marked points with multiplicities. Stacky curves have been around implicitly in the literature for a long time, but the stacky language is sometimes avoided. Some notable applications of stacky curves are: Darmon 1997 uses stacky curves under the name of M-curves to upgrade Faltings Theorem and give applications to the generalized Fermat equation \(x^p + y^q = z^r\); Abramovich and Vistoli 2002 use stacky curves to define a completion of the moduli stack of stable maps into a Deligne-Mumford stack; and Voight and Zureick-Brown 2022 consider modular curves as stacky curves to compute the number of generators and relations for rings of modular forms.
A central motivation to study vector bundles on stacky curves is to provide a cleaner alternative to Mehta and Seshadri's theory of parabolic bundles. Their main idea is to relate representations of the fundamental group of punctured Riemann surfaces to vector bundles on the closed surface by introducing "parabolic structures" at the punctures, i.e. a flag and a real number for each step in the flag. In some sense stacky curves sit in between the punctured surface and the closed surface, allowing us to approximate parabolic bundles by vector bundles on stacky curves. Concretely, moduli spaces of parabolic bundles are isomorphic to moduli spaces of vector bundles on stacky curves.
Current work
My PhD thesis consists of the following three projects.
Uniformisation of spherical stacky curves
Stacky curves come with a genus, which is a rational number \(g \geq 0\). Classically curves fall into three groups, the spherical curves with \(g = 0\), flat curves with \(g = 1\) and hyperbolic curves with \(g \geq 2\), which all exhibit very different behaviour and are often studied separately. A similar trichotomy exists for stacky curves, where we should consider the spherical case \(g < 1\), the flat case \(g = 1\) and the hyperbolic case \(g > 1\). The literature on hyperbolic stacky curves is vast, on the other hand, the spherical case is only really considered over the complex numbers, as for example by Behrend and Noohi 2006. This might not seem like a problem since classically the spherical curves are not particularly interesting, but in the stacky case there are many more spherical curves.
For a spherical stacky curve over an arbitrary field, I provide a minimal field extension over which it can be explicitly uniformised. As a consequence I can state a McKay-type correspondence between stacky curves and the finite Dynkin diagrams. This generalizes a result of Geigle and Lenzing 1987, which works over algebraically closed fields and thus only observes the simply laced Dynkin diagrams. Another application is to classify the finite subgroups of \(\operatorname{PGL}_2(k)\) for an arbitrary field \(k\), by carefully analysing the map sending a group \(G \subset \operatorname{PGL}_2(k)\) to the stacky curve \([\mathbb{P}_k^1/G]\). This gives an alternative geometric proof of the results of Beauville 2010.
Formulas for motives of stacks of sheaves on stacky curves
There is a long history of computing invariants of moduli spaces of vector bundles on a curve and closely related moduli spaces. Initially Harder and Narasimhan 1974/75 compute \(\ell\)-adic cohomology groups by point counting over finite fields. Later Atiyah and Bott 1983 give a differential geometric computation of the singular cohomology. Recently Hoskins and Pepin Lehalleur 2021 prove a formula for the Voevodsky motive of the stack of vector bundles on a curve, using moduli theoretic techniques.
I build on this work to show that the motive of the stack of coherent sheaves on a stacky curve is generated by the motive of the curve. Roughly speaking, all (cohomological) invariants of the moduli stack can be expressed in terms of invariants of the curve. The most difficult part is understanding the base case of the stack of torsion sheaves. It turns out that torsion sheaves on stacky curves are closely related to representations of cyclic quivers and along the way we construct a generalized cyclically graded Grothendieck-Springer resolution.
Projectivity of the good moduli space of semistable sheaves on a stacky curve
Alper, Halpern-Leistner, and Heinloth 2023 give general existence criteria for a proper good moduli space of an Artin stack. Alper et al. 2022 verify this existence criterion for the moduli stack of vector bundles on a curve. In addition an ample line bundle is constructed, thus proving the projectivity of the good moduli space of the stack of semistable vector bundles on a curve.
Together with With C. Damiolini, V. Hoskins, S. Makarova we generalize these results to the stack of semistable vector bundles on a stacky curve. On a stacky curve there are many notions of semistability for vector bundles and the construction of our ample line bundle depends strongly on the chosen notion of stability. This work provides a modern proof of projectivity of the moduli space of semistable parabolic bundles on a curve.
Future work
Stability after pushforward
Beauville 2000 conjectured that for a map of curves \(C \rightarrow D\), where the target has genus \(\geq 2\), the generic stable bundle remains stable after pushforward and he proves this for étale morphisms.
It is a general fact that any map of curves factors as \(C \rightarrow \mathcal{X} \rightarrow D\), where \(\mathcal{X}\) is a stacky curve, \(C \rightarrow \mathcal{X}\) is étale and \(\mathcal{X} \rightarrow D\) is a degree \(1\) ramified morphism. We can show that for a specific notion of stability the generic bundle remains stable when pushing forward along \(C \rightarrow \mathcal{X}\) and, for a different notion of stability, a bundle is stable on \(\mathcal{X}\) if and only if it is when pushing forward to \(D\). Together with D. Weissman we intend to use the theory of variation of stability to relate these two different notions of stability and prove Beauville's conjecture.
A geometric description of Campana points
Campana points have been used to interpolate between rational and integral points of varieties and there are several far reaching conjectures on the density of these points. So far the definition of Campana points has been somewhat ad hoc, in the sense that there was no geometric object that these Campana points are points of.
I define a certain Artin stack for which the points are (up to a controlled finite amount of data) exactly the Campana points. I propose to develop a geometric theory of these stacks, at least in the one-dimensional case, with the end goal of attacking some of these conjectures.