arXiv subjects
Publications and source records attributed to Luca Benzo.
We prove that for all integers $r \geq 2$ and $g \geq \lfloor \frac{r^2+10r+1}{4} \rfloor$ there exists a component of the locus $\mathcal{S}^r_g$ of spin curves with a theta characteristic $L$ such that $h^0(L) \geq r+1$ and $h^0(L)\equiv r+1 (\text{mod} 2)$ which has expected codimension $\binom{r+1}{2}$ inside the moduli space $\mathcal{S}_g$ of spin curves of genus $g$.
Let $C$ be a smooth projective curve of genus $g\geq 2$. Fix an integer $r\geq 0$, and let $\underline{k}=(k_1,\ldots,k_n)$ be a sequence of positive integers with $k_1+\ldots+k_n=g-1$. We study $n$-pointed curves $(C,p_1,\ldots,p_n)$ such that the line bundle $L:=O_C\left(\sum_{i=1}^n k_i p_i\right)$ is a theta-characteristic such that $h^0\left(C,L\right)$ is at least $r+1$ and it has the same parity as $r+1$. We prove that they describe a sublocus $\mathcal{G}^r_g(\underline{k})$ of $\mathcal{M}_{g,n}$ having codimension at most $g-1+\frac{r(r-1)}{2}$. Moreover, for any $r\geq 0$, $\underline{k}$ as above, and $g$ greater than an explicit integer $g(r)$ depending on $r$, we present irreducible components of $\mathcal{G}^r_g(\underline{k})$ attaining the maximal codimension in $\mathcal{M}_{g,n}$, so that the bound turns out to be sharp.
Let $(S,L)$ be a smooth primitively polarized K3 surface of genus $g$ and $f:X \rightarrow \mathbb{P}^1$ the fibration defined by a linear pencil in $|L|$. For $f$ general and $g \geq 7$, we work out the splitting type of the locally free sheaf $Ψ^{*}_f T_{\overline{M}_g}$, where $Ψ_f$ is the modular morphism associated to $f$. We show that this splitting type encodes the fundamental geometrical information attached to Mukai's projection map $\mathcal{P}_g \rightarrow \overline{\mathcal{M}}_g$, where $\mathcal{P}_g$ is the stack parameterizing pairs $(S,C)$ with $(S,L)$ as above and $C \in |L|$ a stable curve. Moreover, we work out conditions on a fibration $f$ to induce a modular morphism $Ψ_f$ such that the normal sheaf $N_{Ψ_f}$ is locally free.
Let V^{r}_{d,g, δ} be the Hilbert scheme of nodal curves in P^r of degree d and arithmetic genus g with δnodes. Under suitable numerical assumptions on d and g, for every 0 \le δ\le g we construct an irreducible component of V^{r}_{d,g, δ} having the expected number of moduli.
We prove uniruledness of some moduli spaces $\bar{M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points using linear systems on nonsingular projective surfaces containing the general curve of genus $g$. Precisely we show that $\bar{M}_{g,n}$ is uniruled for $g=12$ and $n \leq 5$, $g=13$ and $n \leq 3$, $g=15$ and $n \leq 2$. We then prove that the pointed hyperelliptic locus $H_{g,n}$ is uniruled for $g \geq 2$ and $n \leq 4g+4$. In the last part we show that a nonsingular complete intersection surface does not carry a linear system containing the general curve of genus $g \geq 16$ and if it carries a linear system containing the general curve of genus $12 \leq g \leq 15$ then it is canonical.
We prove a conjecture of Gavril Farkas claiming that for all integers r \geq 2 and g \geq \binom{r+2}{2} there exists a component of the locus \mathcal{S}^r_g of spin curves with a theta characteristic L such that h^0(L) \geq r+1 and h^0(L)\equiv r+1 (mod 2) which has codimension \binom{r+1}{2} inside the moduli space \mathcal{S}_g of spin curves of genus g.