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1. Monodromy of Kodaira fibrations of genus 3

   https://doi.org/10.1002/mana.202000189  

   Flapan, Laure (November 2022, Mathematische Nachrichten)

   Abstract A Kodaira fibration is a non‐isotrivial fibration from a smooth algebraic surfaceSto a smooth algebraic curveBsuch that all fibers are smooth algebraic curves of genusg. Such fibrations arise as complete curves inside the moduli space of genusgalgebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of parametrizing curves whose Jacobians have extra endomorphisms. 

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2. 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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3. Biased $$2 \times 2$$ periodic Aztec diamond and an elliptic curve

   https://doi.org/10.1007/s00440-023-01195-8  

   Borodin, Alexei; Duits, Maurice (February 2023, Probability Theory and Related Fields)

   Abstract We study random domino tilings of the Aztec diamond with a biased$$2 \times 2$$ $2\times 2$periodic weight function and associate a linear flow on an elliptic curve to this model. Our main result is a double integral formula for the correlation kernel, in which the integrand is expressed in terms of this flow. For special choices of parameters the flow is periodic, and this allows us to perform a saddle point analysis for the correlation kernel. In these cases we compute the local correlations in the smooth disordered (or gaseous) region. The special example in which the flow has period six is worked out in more detail, and we show that in that case the boundary of the rough disordered region is an algebraic curve of degree eight. 

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4. OBSTRUCTIONS TO ALGEBRAIZING TOPOLOGICAL VECTOR BUNDLES

   https://doi.org/10.1017/fms.2018.16  

   ASOK, A.; FASEL, J.; HOPKINS, M. (March 2019, Forum of mathematics)

   Suppose is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, that is, lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension , it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension that algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is nontrivial in general. 

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5. The essential dimension of congruence covers

   https://doi.org/10.1112/S0010437X21007594  

   Farb, Benson; Kisin, Mark; Wolfson, Jesse (November 2021, Compositio Mathematica)

   Abstract Consider the algebraic function $$\Phi _{g,n}$$ that assigns to a general $$g$$ -dimensional abelian variety an $$n$$ -torsion point. A question first posed by Klein asks: What is the minimal $$d$$ such that, after a rational change of variables, the function $$\Phi _{g,n}$$ can be written as an algebraic function of $$d$$ variables? Using techniques from the deformation theory of $$p$$ -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $$p$$ -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $$p$$ -dimension of congruence covers of the moduli space of genus $$g$$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $$M$$ is proper . As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety. 

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