Non-tautological Hurwitz cycles

We show that various loci of stable curves of sufficiently large genus admitting degree d covers of positive genus curves define non-tautological algebraic cycles on $${\overline{{\mathcal {M}}}}_{g,N}$$ M ¯ g , N , assuming the non-vanishing of the d-th Fourier coefficient of a certain modular form. Our results build on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by Schmitt-van Zelm and the author.

The $${\mathcal {H}}$$-tautological ring

We extend the theory of tautological classes on moduli spaces of stable curves to the more general setting of moduli spaces of admissible Galois covers of curves, introducing the so-called $${\mathcal {H}}$$ H -tautological ring. The main new feature is the existence of restriction-corestriction morphisms remembering intermediate quotients of Galois covers, which are a rich source of new classes. In particular, our new framework includes classes of Harris–Mumford admissible covers on moduli spaces of curves, which are known in some (and speculatively many more) examples to lie outside the usual tautological ring. We give additive generators for the $${\mathcal {H}}$$ H -tautological ring and show that their intersections may be algorithmically computed, building on work of Schmitt-van Zelm. As an application, we give a method for computing integrals of Harris-Mumford loci against tautological classes of complementary dimension, recovering and giving a mild generalization of a recent quasi-modularity result of the author for covers of elliptic curves.

Fibonacci, Golden Ratio, and Vector Bundles

There is a family of vector bundles over the moduli space of stable curves that, while first appearing in theoretical physics, has been an active topic of study for algebraic geometers since the 1990s. By computing the rank of the exceptional Lie algebra g2 case of these bundles in three different ways, a family of summation formulas for Fibonacci numbers in terms of the golden ratio is derived.

The Dual Complex of a Semi-log Canonical Surface

Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary $\Delta $ of a log canonical pair $(X,\Delta )$ and also appear as limits of canonically polarized varieties in moduli theory. For certain three-fold pairs $(X,\Delta ),$ we show how to compute the PL homeomorphism type of the dual complex of a dlt minimal model directly from the normalization data of $\Delta $.

Birational models of 𝓜2,2 arising as moduli of curves with nonspecial divisors

We study birational projective models of 𝓜2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜 2,2(𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M 2,2(𝓩).

Topology of tropical moduli of weighted stable curves

The moduli space Δg,w of tropical w-weighted stable curves of volume 1 is naturally identified with the dual complex of the divisor of singular curves in Hassett’s spaces of w-weighted stable curves. If at least two of the weights are 1, we prove that Δ0, w is homotopic to a wedge sum of spheres, possibly of varying dimensions. Under additional natural hypotheses on the weight vector, we establish explicit formulas for the Betti numbers of the spaces. We exhibit infinite families of weights for which the space Δ0,w is disconnected and for which the fundamental group of Δ0,w has torsion. In the latter case, the universal cover is shown to have a natural modular interpretation. This places the weighted variant of the space in stark contrast to the heavy/light cases studied previously by Vogtmann and Cavalieri–Hampe–Markwig–Ranganathan. Finally, we prove a structural result relating the spaces of weighted stable curves in genus 0 and 1, and leverage this to extend several of our genus 0 results to the spaces Δ1,w.