arXiv subjects
Publications and source records attributed to Dario Faro.
Let $S$ be a very general Enriques surface. In this paper we show that $Ω_{S}|_C$ is semistable for a general curve $C \in |H|$, where $H$ is an effective line bundle on $S$. Moreover, we give explicit constructions of families of smooth irreducible curves that destabilize $Ω_S$
Finiteness and injectivity of the moduli morphisms $μ_n\colon|nH|_{\rm{sm}}\to\mathcal{M}_{g_n}$ 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 multiple is sufficiently positive, so our focus is on low-degree phenomena. In particular, for a general K3 surface of degree $2$, we prove that the primitive moduli map is quasi-finite but ramified at exactly $171$ points, identified with the jumping lines of a logarithmic tangent bundle associated with the branch sextic. On the other hand, we prove that if $X\subset\mathbb{P}^n$ is a general hypersurface of degree at least $4$, then $\rm{H}^0(Y,T_X|_Y)=0$ for every smooth hyperplane section $Y$ of $X$. In particular, the moduli map for smooth hyperplane sections of a general quartic K3 surface is unramified. We moreover show that ramification does occur for higher multiples of the primitive polarisation on both degree-$2$ and degree-$4$ K3 surfaces, and relate the question to the stability of the tangent bundle.
In this paper we study higher even Gaussian maps of the canonical bundle on hyperelliptic curves and we determine their rank, giving explicit descriptions of their kernels. Then we use this descriptions to investigate the hyperelliptic Torelli map $j_h$ and its second fundamental form. We study isotropic subspaces of the tangent space $T_{{\mathcal H}_g, [C]}$ to the moduli space ${\mathcal H}_g$ of hyperelliptic curves of genus $g$ at a point $[C]$, with respect to the second fundamental form $ρ_{HE}$ of $j_h$. In particular, for any Weierstrass point $p \in C$, we construct a subspace $V_p$ of dimension $\lfloor\frac{g}{2} \rfloor$ of $T_{{\mathcal H}_g, [C]}$ generated by higher Schiffer variations at $p$, such that the only isotropic tangent direction $ζ\in V_p$ for the image of $ρ_{HE}$ is the standard Schiffer variation $ξ_p$ at the Weierstrass point $p \in C$.
In this paper we study higher Gaussian (or Wahl) maps for the canonical bundle of certain smooth projective curves. More precisely, we determine the rank of higher Gaussian maps of the canonical bundle for plane curves, for curves contained in certain linear systems in a surface given by a product of two curves and for curves contained in a sufficiently ample line bundle on an Enriques surface.
In recent work by Arena, Canning, Clader, Haburcak, Li, Mok, and Tamborini it was proven that for infinitely many values of $g$ and $n$, there exist non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ of smooth genus $g$, $n$-pointed curves. Here we show how a generalization of their technique allows to cover most of the remaining cases, proving the existence of non-tautological algebraic cohomology classes on the moduli space $\mathcal{M}_{g,n}$ for all but finitely many values of $g$ and $n$.
We prove that the $k$-th Gaussian map $γ^k_{H}$ is surjective on a polarized unnodal Enriques surface $(S, H)$ with $ϕ(H)>2k+4$. In particular, as a consequence, when $ϕ(H)>4(k+2)$, we obtain the surjectivity of the $k$-th Gauss-Prym map $γ^k_{ω_C\otimesα}$ on smooth hyperplane sections $C\in \vert H\vert.$ In case $k=1$ it is sufficient to ask $ϕ(H)>6$.
We give obstructions - in terms of Gaussian maps - for a marked Prym curve $(C,α,T_d)$ to admit a singular model lying on an Enriques surface with only one $d$-ordinary point singularity and in such a way that $T_d$ corresponds to the divisor over the singular point.