Let $\ell \ge 7$ be a prime, $C$ be the non-singular projective curve defined over $\mathbb{Q}$ by the affine model $y(1-y)=x^{\ell }$, $\infty$ the point of $C$ at infinity on this model, $J$ the Jacobian of $C$, and $\phi :C\to J$ the albanese embedding with $\infty$ as base point. Let $\overline{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$. Taking care of a case not covered in [12], we show that $\phi \left(C\right)\cap J_{tors}\left(\overline{\mathbb{Q}}\right)$ consists only of the image under $\phi$ of the Weierstrass points of $C$ and the points $(x,y)=(0,0)$ and $(0,1)$, where $J_{tors}$ denotes the torsion points of $J$.

We study the integral model of the Drinfeld modular curve $X_1\left(n\right)$ for a prime $n\in {𝔽}_q\left[T\right]$. A function field analogue of the theory of Igusa curves is introduced to describe its reduction mod $n$. A result describing the universal deformation ring of a pair consisting of a supersingular Drinfeld module and a point of order $n$ in terms of the Hasse invariant of that Drinfeld module is proved. We then apply Jung-Hirzebruch resolution for arithmetic surfaces to produce a regular model of $X_1\left(n\right)$ which, after contractions in...

In questa nota si dà una descrizione della fibra della mappa di Prym in genere 4. Se $JX$ è la Jacobiana di una curva di genere 3, allora la fibra della mappa di Prym in $JX$ si ottiene dalla varietà di Kummer $KX$ mediante due scoppiamenti: ${\sigma }_1:KX\left(0\right)\to KX$ che è lo scoppiamento di $KX$ nell'origine e ${\sigma }_2:\widetilde{KX\left(0\right)}\to KX\left(0\right)$ che è lo scoppiamento lungo una curva isomorfa a $X$.

For any prime number p > 3 we compute the formal completion of the Néron model of J0(p) in terms of the action of the Hecke algebra on the Z-module of all cusp forms (of weight 2 with respect to Γ0(p)) with integral Fourier development at infinity.

We study the enumerative geometry of the moduli space ${ℛ}_g$ of Prym varieties of dimension $g-1$. Our main result is that the compactication of ${ℛ}_g$ is of general type as soon as $g>13$ and $g$ is different from 15. We achieve this by computing the class of two types of cycles on ${ℛ}_g$: one defined in terms of Koszul cohomology of Prym curves, the other defined in terms of Raynaud theta divisors associated to certain vector bundles on curves. We formulate a Prym–Green conjecture on syzygies of Prym-canonical curves....

We study a moduli space ${\mathrm{𝒜𝒮}}_g$ for Artin-Schreier curves of genus $g$ over an algebraically closed field $k$ of characteristic $p$. We study the stratification of ${\mathrm{𝒜𝒮}}_g$ by $p$-rank into strata ${\mathrm{𝒜𝒮}}_{g.s}$ of Artin-Schreier curves of genus $g$ with $p$-rank exactly $s$. We enumerate the irreducible components of ${\mathrm{𝒜𝒮}}_{g,s}$ and find their dimensions. As an application, when $p=2$, we prove that every irreducible component of the moduli space of hyperelliptic $k$-curves with genus $g$ and $2$-rank $s$ has dimension $g-1+s$. We also determine all pairs $(p,g)$ for...