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1. Counting points on smooth plane quartics

   https://doi.org/10.1007/s40993-022-00397-8  

   Costa, Edgar; Harvey, David; Sutherland, Andrew V. (March 2023, Research in Number Theory)

   Abstract We present efficient algorithms for counting points on a smooth plane quartic curve X modulo a prime p . We address both the case where X is defined over  $${\mathbb {F}}_p$$ F p and the case where X is defined over $${\mathbb {Q}}$$ Q and p is a prime of good reduction. We consider two approaches for computing $$\#X({\mathbb {F}}_p)$$ # X ( F p ) , one which runs in $$O(p\log p\log \log p)$$ O ( p log p log log p ) time using $$O(\log p)$$ O ( log p ) space and one which runs in $$O(p^{1/2}\log ^2p)$$ O ( p 1 / 2 log 2 p ) time using $$O(p^{1/2}\log p)$$ O ( p 1 / 2 log p ) space. Both approaches yield algorithms that are faster in practice than existing methods. We also present average polynomial-time algorithms for $$X/{\mathbb {Q}}$$ X / Q that compute $$\#X({\mathbb {F}}_p)$$ # X ( F p ) for good primes $$p\leqslant N$$ p ⩽ N in $$O(N\log ^3 N)$$ O ( N log 3 N ) time using O ( N ) space. These are the first practical implementations of average polynomial-time algorithms for curves that are not cyclic covers of $${\mathbb {P}}^1$$ P 1 , which in combination with previous results addresses all curves of genus $$g\leqslant 3$$ g ⩽ 3 . Our algorithms also compute Cartier–Manin/Hasse–Witt matrices that may be of independent interest. 

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2. Compactifications of Moduli of Points and Lines in the Projective Plane

   https://doi.org/10.1093/imrn/rnab200  

   Schaffler, Luca; Tevelev, Jenia (August 2021, International Mathematics Research Notices)

   Abstract Projective duality identifies the moduli spaces $$\textbf{B}_n$$ and $$\textbf{X}(3,n)$$ parametrizing linearly general configurations of $$n$$ points in $$\mathbb{P}^2$$ and $$n$$ lines in the dual $$\mathbb{P}^2$$, respectively. The space $$\textbf{X}(3,n)$$ admits Kapranov’s Chow quotient compactification $$\overline{\textbf{X}}(3,n)$$, studied also by Lafforgue, Hacking, Keel, Tevelev, and Alexeev, which gives an example of a KSBA moduli space of stable surfaces: it carries a family of certain reducible degenerations of $$\mathbb{P}^2$$ with $$n$$ “broken lines”. Gerritzen and Piwek proposed a dual perspective, a compact moduli space parametrizing certain reducible degenerations of $$\mathbb{P}^2$$ with $$n$$ smooth points. We investigate the relation between these approaches, answering a question of Kapranov from 2003. 

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3. Shafarevich’s Conjecture for Families of Hypersurfaces over Function Fields

   https://doi.org/10.1093/imrn/rnaf225  

   Engel, Philip; Lin, Alice; Tayou, Salim (August 2025, International Mathematics Research Notices)

   Given a smooth quasi-projective complex algebraic variety $$\mathcal{S}$$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $$\mathcal{S}$$ of degree $$d$$ in $$\mathbb{P}_{\mathbb C}^{n+1}$$. We prove that the finiteness is uniform in $$\mathcal{S}$$ and give examples where the result is sharp. We also prove similar results for certain complete intersections in $$\mathbb{P}_{\mathbb C}^{n+1}$$ of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem. 

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4. Hyperbolic Metric Spaces and Stochastic Embeddings

   https://doi.org/10.1017/fms.2024.118  

   Gartland, Chris (February 2025, Forum of Mathematics, Sigma)

   Abstract Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, we build out some of the basic theory of stochastic embeddings in the infinite setting with an aim toward applications to Lipschitz free space theory. We prove that proper metric spaces stochastically embedding into$$\mathbb {R}$$-trees have Lipschitz free spaces isomorphic to$$L^1$$-spaces. We then undergo a systematic study of stochastic embeddability of Gromov hyperbolic metric spaces into$$\mathbb {R}$$-trees by way of stochastic embeddability of their boundaries into ultrametric spaces. The following are obtained as our main results: (1) every snowflake of a compact, finite Nagata-dimensional metric space stochastically embeds into an ultrametric space and has Lipschitz free space isomorphic to$$\ell ^1$$, (2) the Lipschitz free space over hyperbolicn-space is isomorphic to the Lipschitz free space over Euclideann-space and (3) every infinite, finitely generated hyperbolic group stochastically embeds into an$$\mathbb {R}$$-tree, has Lipschitz free space isomorphic to$$\ell ^1$$, and admits a proper, uniformly Lipschitz affine action on$$\ell ^1$$. 

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5. Realizing trees of configurations in thin sets

   https://doi.org/10.2140/pjm.2025.335.355  

   Greenleaf, Allan; Iosevich, Alex; Taylor, Krystal (January 2025, Pacific Journal of Mathematics)

   Let $$\phi(x,y)$$ be a continuous function, smooth away from the diagonal, such that, for some $$\alpha>0$$, the associated generalized Radon transforms \begin{equation} \label{Radon} R_t^{\phi}f(x)=\int_{\phi(x,y)=t} f(y) \psi(y) d\sigma_{x,t}(y) \end{equation} map $$L^2({\mathbb R}^d) \to H^{\alpha}({\mathbb R}^d)$$ for all $t>0$. Let $$E$$ be a compact subset of $${\mathbb R}^d$$ for some $$d \ge 2$$, and suppose that the Hausdorff dimension of $$E$$ is $$>d-\alpha$$. We show that any tree graph $$T$$ on $k+1$ ($$k \ge 1$$) vertices is realizable in $$E$$, in the sense that there exist distinct $$x^1, x^2, \dots, x^{k+1} \in E$$ and $t>0$ such that the $$\phi$$-distance $$\phi(x^i, x^j)$$ is equal to $$t$$ for all pairs $(i,j)$ corresponding to the edges of the graph $$T$$. 

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