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1. Bitangents to symmetric quartics

   Bethea, Candace; Brazelton, Thomas (October 2024, arXiv pre-print repository)

   NA (Ed.)

   Recall that a non-singular planar quartic is a canonically embedded non-hyperelliptic curve of genus three. We say such a curve is symmetric if it admits non-trivial automorphisms. The classification of (necessarily finite) groups appearing as automorphism groups of non-singular curves of genus three dates back to the last decade of the 19th century. As these groups act on the quartic via projective linear transformations, they induce symmetries on the 28 bitangents. Given such an automorphism group G=Aut(C), we prove the G-orbits of the bitangents are independent of the choice of C, and we compute them for all twelve types of smooth symmetric planar quartic curves. We further observe that techniques deriving from equivariant homotopy theory directly reveal patterns which are not obvious from a classical moduli perspective. 

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2. Reduction of plane quartics and Dixmier–Ohno invariants

   https://doi.org/10.1007/s40993-024-00590-x  

   van_Bommel, Raymond; Docking, Jordan; Lercier, Reynald; García, Elisa Lorenzo (March 2025, Research in Number Theory)

   Abstract We characterise, in terms of Dixmier–Ohno invariants, the types of singularities that a plane quartic curve can have. We then use these results to obtain new criteria for determining the stable reduction types of non-hyperelliptic curves of genus 3. 

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3. Equivariant Witt complexes and twisted topological Hochschild homology

   https://doi.org/10.1016/j.topol.2025.109444  

   Bohmann, Anna Marie; Gerhardt, Teena; Krulewski, Cameron; Petersen, Sarah; Yang, Lucy (May 2025, Topology and its Applications)

   The topological Hochschild homology of a ring (or ring spectrum) R is an S1-spectrum, and the fixed points of THH(R) for subgroups C_n of S1 have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, drawing on the definition of equivariant Witt vectors by Blumberg, Gerhardt, Hill and Lawson, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra. 

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4. Towards the Sato–Tate groups of trinomial hyperelliptic curves

   https://doi.org/10.1142/S1793042121500822  

   Emory, Melissa; Goodson, Heidi; Peyrot, Alexandre (November 2021, International Journal of Number Theory)

   We consider the identity component of the Sato–Tate group of the Jacobian of curves of the form C1 : y2 = x2g+2 + c, C2 : y2 = x2g+1 + cx, C3 : y2 = x2g+1 + c, where g is the genus of the curve and c in Q* is constant. We approach this problem in three ways. First we use a theorem of Kani-Rosen to determine the splitting of Jacobians for C1 curves of genus 4 and 5 and prove what the identity component of the Sato–Tate group is in each case. We then determine the splitting of Jacobians of higher genus C1 curves by finding maps to lower genus curves and then computing pullbacks of differential 1-forms. In using this method, we are able to relate the Jacobians of curves of the form C1, C2 and C3. Finally, we develop a new method for computing the identity component of the Sato–Tate groups of the Jacobians of the three families of curves. We use this method to compute many explicit examples, and find surprising patterns in the shapes of the identity components ST^0(C) for these families of curves. 

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5. Equivariant double-slice genus, stabilization, and equivariant stabilization

   https://doi.org/10.2140/agt.2025.25.5137  

   Gabbard, Malcolm (January 2025, Algebraic & Geometric Topology)

   We define the equivariant double-slice genus and equivariant superslice genus of a strongly invertible knot. We prove lower bounds for both the equivariant double-slice genus and the equivariant superslice genus. Using these bounds we find a family of knots which are double-slice and equivariantly slice, but have equivariant double-slice genus at least n. Using this result, we construct unknotted symmetric 2-spheres which do not bound symmetric 3-balls. Additionally, using double-slice and superslice genera we find effective lower bounds for 1-handle stabilization distance and identify a possible method for using equivariant double-slice and superslice genera to bound the symmetric 1-handle stabilization distance for symmetric surfaces. 

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