We prove that for a very general abelian variety of dimension $\geq 4$, a divisor $D\in {\rm CH}^1(A)$ that satisfies $D^2=0$ in ${\rm CH}^2(A)$ is of torsion. The same result is also established for a very general Jacobian in genus $4$. We use then the second statement in order to prove a conjecture of Pirola, which states that any rational section of the Kummer fibration $K=J/\pm {\rm Id}\rightarrow \mathcal{M}_4$, where $J\rightarrow \mathcal{M}_4 $ is the Jacobian fibration, must be a multiple of the Griffiths-Pirola section given by the difference of the two trigonal divisors.
Let $H$ be a numerical monoid, that is, a cofinite submonoid of $\mathbb N$ (the non-negative integers under addition). Denote by $\mathcal P_{\text{fin},0}(H)$ the monoid obtained by endowing the family of all finite subsets of $H$ containing $0$ with the operation of setwise addition induced by $H$ on its power set. Tringali and Yan [JCTA, 2025] have recently established that $\mathcal P_{\text{fin},0}(\mathbb N)$ has a unique non-trivial automorphism, and conjectured that the automorphism group of $\mathcal P_{\text{fin},0}(H)$ is trivial whenever $H \ne \mathbb N$. We prove this conjecture and, as a byproduct, give a new proof of the Tringali--Yan theorem.
Let $E/\mathbb Q$ be an elliptic curve and for each prime $p$, let $N_p$ denote the number of points of $E$ modulo $p$. The original version of the conjecture of Birch and Swinnerton-Dyer asserts that $\prod \limits _{p \leq x} \frac{N_p}{p} \sim C (\log x) ^{\text{rank}(E(\mathbb Q))}$ as $x \to \infty$. In this paper, we formulate a similar conjectural asymptotic for smooth projective curves of genus at least 2, in which the contributions to the conjectured asymptotic come not only from the rank of the Jacobian but also from the Sato--Tate group of the curve. The key analytic input in formulating our conjecture is a conjecture due to Kurokawa (2012) on the convergence of Euler products of entire $L$-functions on the critical line. We also provide some numerical evidence for our conjecture in various cases.
Let $G$ be the topological fundamental group of a nonsingular complex projective surface. Troncoso and Urz\'ua proved that the Chern slopes $c_1^2/c_2$ of minimal surfaces of general type $S$ with $\pi_1(S)\simeq G$ are dense in $[1,3]$, and left $[1/3,1)$ open. We prove that they are dense in $[1/2,3]$, an interval that cannot be enlarged without contradicting either a theorem of Mendes Lopes and Pardini or Reid's conjecture. We prove more: the slopes of such surfaces with $K_S$ ample are dense in $[1/2,2]$, the first case in which the conjecture of Troncoso and Urz\'ua on ample canonical classes is established. The tool is an exact ampleness criterion for their product construction, which shows in particular that their own surfaces never have ample canonical class, whatever the defining sections.
Over any infinite field we prove existence of polynomial automorphism with prescribed differentials at points. More precisely, let $\K$ be an infinite field and $a_1,\ldots, a_k;$ $b_1,\ldots , b_k$ be two sequences of different points in~$\K^n$, $n\ge 2$. For any sequence $L_1,\ldots, L_k\in SL(n,\K)$ there exists a polynomial automorphism $\Phi: \K^n\to \K^n$ with jacobian one such that $\Phi(a_i)=b_i$ and $d_{a_i} \Phi= L_i$ for $i=1,\ldots, k.$
Zbigniew Jelonek, Gustavo Menani, Maria Michalska· 0 citations
We show that any $d$-dimensional local ring $A$ with a dualizing complex, $\mathrm{depth} A=d-1$, and cyclic deficiency module $K^{d-1}(A)$ admits a maximal Cohen--Macaulay module. It is constructed as the unique nonzero cohomology module of the cone of the derived morphism induced by a surjection $A\to K^{d-1}(A)$. When $A$ is quasi-Gorenstein, this module is identified with the first syzygy of the canonical module $\omega_{A/xA}$, for any $x\in\operatorname{ann}_A K^{d-1}(A)$ that is regular on $A$. This recovers a theorem of Tavanfar and Shimomoto in the $3$-dimensional quasi-Gorenstein case with $K^2(A)\cong k$. We also give examples of section rings satisfying the hypotheses of our theorem.
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the classical Cauchy and dual Cauchy identities. We pose the problem of giving a direct combinatorial proof of these formulas for $0<k<\min\{m, n\}$. When $k=1$ or $k=\min\{m, n\}-1$, we give such a proof via an explicit sign-reversing involution on certain sets of Littlewood--Richardson tableaux.