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1. A relative orientation for the moduli space of stable maps to a del Pezzo surface.

   Kass, Jesse; Levine, Marc; Solomon, Jake; Wickelgren, Kirsten (December 2026, Algebraic geometry)

   We prove orientation results for evaluation maps of moduli spaces of rational stable maps to del Pezzo surfaces over a field, both in characteristic 0 and in positive characteristic. These results and the theory of degree devel- oped in a sequel produce quadratically enriched counts of rational curves over non-algebraically closed fields of characteristic not 2 or 3. Orientations are constructed in two steps. First, the ramification locus of the evaluation map is shown to be the divisor in the moduli space of stable maps where image curves have a cusp. Second, this divisor is related to the discriminant of a branched cover of the moduli space given generically by pairs of points on the universal curve with the same image. 

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2. On the moduli spaces of 4d N=3 SCFTs I: triple special Kähler structure

   Argyres, Philip; Bouget, Antoine; Martone, Mario (December 2019, ArXivorg)

   We initiate a systematic analysis of moduli spaces of vacua of four dimensional =3 SCFTs. Our analysis is based on the one hand on the properties of =3 chiral rings --- which we review in detail and contrast with chiral rings of theories with less supersymmetry --- and on the other hand on constraints coming from low-energy supersymmetry. This leads us to introduce a new type of geometric structure, which characterizes =3 SCFT moduli spaces, and that we call triple special Kähler (TSK). A rank-n TSK moduli space has complex dimension 3n, and is singular at complex co-dimension 3 subspaces where charged states become massless. The structure of singularities defines a stratification of the TSK space in terms of lower-dimensional TSK manifolds. 

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3. The Rank of the Normal Functions of the Ceresa and Gross–Schoen Cycles

   https://doi.org/10.1017/fms.2025.10089  

   Hain, Richard (January 2025, Forum of Mathematics, Sigma)

   Abstract In this paper we show that the rank of the normal function function of the genus$$g$$Ceresa cycle over the moduli space of curves has the maximal rank possible,$$3g-3$$, provided that$$g\ge 3$$. In genus 3 we show that the Green–Griffiths invariant of this normal function is a Teichmüller modular form of weight$$(4,0,-1)$$and use this to show that the rank of the Ceresa normal function is exactly 1 along the hyperelliptic locus. We also introduce new techniques and tools for studying the behaviour of normal functions along and transverse to boundary divisors. These include the introduction of residual normal functions and the use of global monodromy arguments to compute them. 

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4. On the tensor rank of the 3 x 3 permanent and determinant

   https://doi.org/10.13001/ela.2021.5107  

   Krishna, Siddharth; Makam, Visu (January 2021, The Electronic Journal of Linear Algebra)

   null (Ed.)

   The tensor rank and border rank of the $$3 \times 3$$ determinant tensor are known to be $$5$$ if the characteristic is not two. In characteristic two, the existing proofs of both the upper and lower bounds fail. In this paper, we show that the tensor rank remains $$5$$ for fields of characteristic two as well. 

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5. Tate’s thesis in the de Rham setting

   https://doi.org/10.1090/jams/1010  

   Hilburn, Justin; Raskin, Sam (July 2023, Journal of the American Mathematical Society)

   We calculate the category of D D -modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the 3 d 3d mirror symmetry program, which obtains new compatibilities for the geometric Langlands program from rich dualities of QFTs that are themselves obtained from string theory conjectures. 

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