Talks

I will explain the logarithmic Hilbert scheme associated to a SNC pair $(X,D)$. In particular, I will explain how the valuative criterion for properness of this logarithmic Hilbert scheme leads to an associated object of tropical geometry.

Supplementing Patrick's talk, I will discuss some fundamental aspects of logarithmic Hilbert stacks observed in joint work with Mattia Talpo and Richard Thomas, ultimately using hyperplane arrangements in higher dimensional tropicalizations.

I will discuss recent joint work with Luca Battistella and Dhruv Ranganathan, establishing a comparison between logarithmic and orbifold Gromov-Witten invariants which incorporates negative tangency orders. A key part of our work is the discovery of a "refined virtual class" which gives rise to a distinguished sector of the punctured Gromov-Witten invariants. I will explain how this arises naturally as an intersection product on the Artin fan, and how it can be calculated. I will also sketch the proof of the logarithmic-orbifold comparison theorem.

The Jacobian of the universal family of the moduli space of smooth curves is an abelian scheme whose geometry is deeply intertwined with the enumerative geometry of $\mathcal M_{g,n}$. Extending this interaction to the moduli space of stable curves requires extending the universal Jacobian. Several toroidal compactifications exist, but the interaction works best for the logarithmic Jacobian – a space that is the minimal such compactification but lives purely in the world of logarithmic geometry. In this talk I will explain how to assign a tautological ring to the logarithmic Jacobian – and spaces like the logarithmic Jacobian – , and discuss the problem of calculating the pushforwards of tautological classes to $\overline{\mathcal M}_{g,n}~$, together with connections of this problem to the intersection theory of the moduli space of abelian varieties.

Linear series on smooth curves parameterize, depending on one's perspective, either invertible sheaves and linear subspaces of their global sections, or maps to unparameterized projective spaces. The first perspective was generalized to nodal curves of compact type by Eisenbud and Harris's limit linear series. The second was generalized in rank 1 to nodal curves by Harris and Mumford's admissible covers.

In this talk, I’ll explain how to extend the usual GW/DT correspondence to threefolds with snc boundary. I’ll also discuss logarithmic degeneration formula on both GW and DT sides and prove that it is compatible with the GW/DT conjecture: given an snc degeneration, the correspondence for the generic fiber is implied by the correspondence for the strata of the central fiber. This is joint work with Dhruv Ranganathan.

Logarithmic Gromov-Witten theory and orbifold Gromov-Witten theory offer two distinct paths to probing the enumerative geometry of a normal crossing pair $(X,D)$, and possess complementary strengths and weaknesses. Recent work of Battistella-Nabijou-Ranganathan demonstrates how these two theories are related after appropriate log étale modifications of $(X,D)$. By using known relations in orbifold Gromov-Witten theory, this can be used to produce relations depending on discrete data of maps to $(X,D)$ in the log Gromov-Witten theory of the modification. In the setting where $(X,D)$ is log Calabi-Yau, we show how to use relations in the log Gromov-Witten theory of the modification to deduce relations in the log Gromov-Witten theory of $(X,D)$ which give another proof of associativity of the intrinsic mirror algebra of Gross and Siebert, as well as a proof of the weak Frobenius structure property conjectured by Gross and Siebert.

I will discuss my work (with my collaborators) on degenerations of the stack $\operatorname{Bun}_G(X)$ of $G$-torsors on a smooth projective curve $X$, when the curve degenerates to a nodal curve. The classical McKay correspondence, relating representations of Kleinian groups and ADE singularities, plays a central role in the transition from the case of $\operatorname{GL}_n$ to a general semisimple algebraic group $G$. Higher dimensional Bruhat-Tits group schemes over regular local rings arises as an intrinsic feature in the study; we note that classical Bruhat-Tits theory is only over discrete valuation rings. This higher dimensional analogue can be developed into a full theory. All this can be viewed naturally in the setting of log geometry.

DR cycles (and their various generalisations) might be viewed as the fundamental logGW invariants of toric varieties. For irreducible curves, the DR cycle is a top Chern class of a natural vector bundle, but this fails for more singular curves due to the presence of a certain 'parasitic' subcurve on which non-zero global section can vanish. We show that the genus of the parasitic subcurve controls a certain excess dimension arising in the construction of the DR cycle, yielding (after suitable Riemann-Roch computations) a new proof of Pixton's formula as well as several generalisations. This is joint work with Alessandro Chiodo, see arXiv:2407.09086.

Let $X \to C$ be a projective family of surfaces over a curve with smooth general fibres and simple normal crossing singularity in the special fibre $X_0$. We construct a good compactification of the moduli space of relative length n zero-dimensional subschemes on $X \setminus X_0$ over $C\setminus {0}$. In order to produce this compactification we study expansions of the special fibre $X_0$ together with various GIT stability conditions, generalising the work of Gulbrandsen–Halle–Hulek who use GIT to offer an alternative approach to the work of Li–Wu for Hilbert schemes of points on simple degenerations. We construct stacks equivalent to the underlying stack of some choices of logarithmic Hilbert schemes produced by Maulik–Ranganathan. These methods yield good type III degenerations of Hilbert Schemes of points on K3 surfaces.

In this talk I will give an overview of our naive approach to tropical principal bundles via (extended) affine Weyl groups. This leads to a beautiful and relatively elementary geometric story on the tropical side, which includes analogues of the Weil–Riemann–Roch theorem, the Birkhoff–Grothendieck theorem, Atiyah's classification of vector bundles on elliptic curves, and the Narasimhan–Seshadri correspondence. Much to our surprise this story is sufficient to understand the tropical geometry of the moduli space of semistable vector/principal bundles on the Tate curve and of certain components of the moduli space of semistable vector bundles on maximally degenerate abelian varieties – namely those parametrizing semihomogeneous vector bundles. The pint of magic that makes this possible is Atiyah's classification of vector bundles on elliptic curves and, more generally, the Fourier–Mukai transform on abelian varieties. Given time, I will indulge in some speculations concerning the shape of a general tropical theory of vector bundles (putting into context the recent approach by Khan–Maclagan).