In this paper, we study the degree of irrationality of properly elliptic surfaces with a section. We prove $\min\{\chi(\mathcal O_S),\,2\operatorname{gon}(C)\} \leq \operatorname{irr}(S) \leq 2\operatorname{gon}(C)$. The lower bound is obtained from the canonical bundle formula and the Cayley--Bacharach property. We show that this bound is sharp. We also study the behavior of the degree of irrationality in moduli. A very general properly elliptic surface with a section over a curve of genus at least two has degree of irrationality at least four, whereas special families with $\chi(\mathcal O_S)=1$ or $2$ have degree two. Finally, we investigate properly elliptic surfaces with $\chi(\mathcal O_S)=0$, proving a generic lower bound of four and showing that $\operatorname{irr}(C\times E)=4$ for every hyperelliptic curve $C$ of genus at least two and every elliptic curve $E$. Our paper also includes special properly elliptic surfaces without a section. For Dolgachev surfaces, we exclude degree two for a very general member and construct special examples of degrees two and three.
In this short paper, we prove that, for any two complex elliptic curves \(E\) and \(F\), the product \(E\times F\) has degree of irrationality \(3\). The upper bound is obtained from compatible cyclic plane-cubic models of \(E\) and \(F\): three diagonal bihomogeneous sections define a dominant rational map \(E\times F...
In this second chapter of the $\textit{Arithmetic of critical values}$ series (ACV), we study certain double covers $E_f\to\mathbb{P}^1$ whose branch locus coincides with that of a quartic polynomial $f$. We give a direct proof of the fact, already shown non-constructively in ACV I, that the elliptic curves $E_f$ admit...
Let $k$ be an algebraically closed field of characteristic $2$ and let $q=2^h$, $h\ge3$. We construct ordinary bielliptic curves $X$ of genus $q+1$ for which \[ \Aut(X)\cong \Dih(C_q)\times C_2, \qquad |\Aut(X)|=4q=4(g(X)-1). \] These curves realize case \textup{(ib)} in the classification of Giulietti--Korchm\'aros an...
We prove a conjecture proposed by the first-named author in 1998. Let $f\colon\mathbb{C}\to\mathbb{P}^n$ be a transcendental linearly non-degenerate holomorphic curve of finite lower order. If the counting function $N_1(r,f)$ of its Wronskian zeros satisfies $N_1(r,f)=o(T(r,f))$, then its order and lower order coincide...
The study of heavenly abelian varieties is motivated by a question of Ihara. When an elliptic curve $E/K$ is heavenly at $\ell$, the extension $K(E[\ell^\infty])/K(\mu_\ell^\infty)$ is pro-$\ell$ and unramified away from $\ell$. These are the same arithmetic conditions as the fixed field of the kernel attached to pro-$...
Finiteness and injectivity of the moduli morphisms $\mu_n\colon|nH|_{\rm{sm}}\to\mathcal{M}_{g_n}$ associated with linear systems on K3 surfaces are understood in many cases. In this paper we study their ramification. Existing vanishing and stability results already imply unramifiedness when the polarisation or the mul...
Dario Faro, Frank Gounelas· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
