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On the locus of Prym curves where the Prym--canonical map is not an embedding

Ciliberto Ciro, Dedieu Thomas, Galati Concettina, Knutsen Andreas Leopold
13 Mar 2019 math.AG arxiv.org/abs/1903.05702

We prove that the locus of Prym curves $(C,\eta)$ of genus $g \geq 5$ for which the Prym-canonical system $|\omega_C(\eta)|$ is base point free but the Prym--canonical map is not an embedding is irreducible and unirational of dimension $2g+1$.

Moduli of curves on Enriques surfaces

Ciliberto Ciro, Dedieu Thomas, Galati Concettina, Knutsen Andreas Leopold
19 Feb 2019 math.AG arxiv.org/abs/1902.07142

We compute the number of moduli of all irreducible components of the moduli space of smooth curves on Enriques surfaces. In most cases, the moduli maps to the moduli space of Prym curves are generically injective or dominant. Exceptional behaviour is related to existence of Enriques--Fano threefolds and to curves with nodal Prym-canonical model.

Irreducible unirational and uniruled components of moduli spaces of polarized Enriques surfaces

Ciliberto Ciro, Dedieu Thomas, Galati Concettina, Knutsen Andreas Leopold
27 Sep 2018 math.AG arxiv.org/abs/1809.10569

We give an explicit description of the irreducible components of the moduli spaces of polarized Enriques surfaces in terms of decompositions of the polarization as an effective sum of isotropic classes. We prove that infinitely many of these components are unirational (resp. uniruled). In particular, this applies to components of arbitrarily large genus $g$ and $\phi$-invariant of the polarization.

Moduli of nodal curves on K3 surfaces

Ciliberto Ciro, Flamini Flaminio, Galati Concettina, Knutsen Andreas Leopold
25 Feb 2015 math.AG arxiv.org/abs/1502.07378

We consider modular properties of nodal curves on general $K3$ surfaces. Let $\mathcal{K}p$ be the moduli space of primitively polarized $K3$ surfaces $(S,L)$ of genus $p\geqslant 3$ and $\mathcal{V}{p,m,\delta}\to \mathcal{K}p$ be the universal Severi variety of $\delta$--nodal irreducible curves in $|mL|$ on $(S,L)\in \mathcal{K}_p$. We find conditions on $p, m,\delta$ for the existence of an irreducible component $\mathcal{V}$ of $\mathcal{V}{p,m,\delta}$ on which the moduli map $\psi: \mathcal{V}\to \mathcal{M}_g$ (with $g= m^2 (p -1) + 1-\delta$) has generically maximal rank differential. Our results, which for any $p$ leave only finitely many cases unsolved and are optimal for $m\geqslant 5$ (except for very low values of $p$), are summarized in Theorem 1.1 in the introduction.

Seshadri constants of $K3$ surfaces of degrees 6 and 8

Galati Concettina, Knutsen Andreas Leopold
14 May 2012 math.AG arxiv.org/abs/1205.2982

We compute Seshadri constants $\eps(X):= \eps(\O_X(1))$ on $K3$ surfaces $X$ of degrees 6 and 8. Moreover, more generally, we prove that if $X$ is any embedded $K3$ surface of degree $2r-2 \geq 8$ in $\PP^r$ not containing lines, then $1 < \eps(X) <2$ if and only if the homogeneous ideal of $X$ is not generated by only quadrics (in which case $\eps(X)=3/2$).

On the existence of curves with a triple point on a K3 surface

Galati Concettina
13 Nov 2011 math.AG arxiv.org/abs/1111.3051

Let $(S,H)$ be a general primitively polarized $K3$ surface of genus $\p$ and let $p_a(nH)$ be the arithmetic genus of $nH.$ We prove the existence in $|\mathcal O_S(nH)|$ of curves with a triple point and $A_k$-singularities. In particular, we show the existence of curves of geometric genus $g$ in $|\mathcal O_S(nH)|$ with a triple point and nodes as singularities and corresponding to regular points of their equisingular deformation locus, for every $1\leq g\leq p_a(nH)-3$ and $(\p,n)\neq (4,1).$ Our result is obtained by studying the versal deformation space of a non-planar quadruple point.

On the existence of curves with $A_k$-singularities on $K3$-surfaces

Galati Concettina, Knutsen Andreas Leopold
22 Jul 2011 math.AG arxiv.org/abs/1107.4568

Let $(S,H)$ be a general primitively polarized $K3$ surface. We prove the existence of curves in $|\mathcal O_S(nH)|$ with $A_k$-singularities and corresponding to regular points of the equisingular deformation locus. Our result is optimal for $n=1$. As a corollary, we get the existence of elliptic curves in $|\mathcal O_S(nH)|$ with a cusp and nodes or a simple tacnode and nodes. We obtain our result by studying the versal deformation family of the $m$-tacnode. Finally, we give a regularity condition for families of curves with only $A_k$-singularities in $|\mathcal O_S(nH)|.$

Degenerating curves and surfaces: first results

Galati Concettina
01 Oct 2008 math.AG arxiv.org/abs/0810.0105

Let $\mathcal S\to\mathbb A^1$ be a smooth family of surfaces whose general fibre is a smooth surface of $\mathbb P^3$ and whose special fibre has two smooth components, intersecting transversally along a smooth curve $R$. We consider the Universal Severi-Enriques variety $\mathcal V$ on $\mathcal S\to\mathbb A^1$. The general fibre of $\mathcal V$ is the variety of curves on $\mathcal S_t$ in the linear system $|\mathcal O_{\mathcal S_t}(n)|$ with $k$ cusps and $\delta$ nodes as singularities. Our problem is to find all irreducible components of the special fibre of $\mathcal V$. In this paper, we consider only the cases $(k,\delta)=(0,1)$ and $(k,\delta)=(1,0)$. In particular, we determine all singular curves on the special fibre of $\mathcal S$ which, counted with the right multiplicity, are a limit of 1-cuspidal curves on the general fibre of $\mathcal S$.

Number of moduli of irreducible families of plane curves with nodes and cusps

Galati Concettina
04 Apr 2007 math.AG arxiv.org/abs/0704.0618

Consider the family S of irreducible plane curves of degree n with d nodes and k cusps as singularities. Let W be an irreducible component of S. We consider the natural rational map from W to the moduli space of curves of genus g=(n-1)(n-2)/2-d-k. We define the "number of moduli of W" as the dimension of the image of W with respect to this map. If W has the expected dimension equal to 3n+g-1-k, then the number of moduli of W is at most equal to the min(3g-3, 3g-3+\rho-k), dove \rho is the Brill-Neother number of the linear series of degree n and dimension 2 on a smooth curve of genus g. We say that W has the expected number of moduli if the equality holds. In this paper we construct examples of families of irreducible plane curves with nodes and cusps as singularities having expected number of moduli and with non-positive Brill-Noether number.

On the number of moduli of plane sextics with six cusps

Galati Concettina
04 Apr 2007 math.AG arxiv.org/abs/0704.0622

Let S be the variety of irreducible sextics with six cusps as singularities. Let W be one of irreducible components of W. Denoting by M_4 the space of moduli of smooth curves of genus 4, the moduli map of W is the rational map from W to M_4 sending the general point of W, corresponding to a plane curve D, to the point of M_4 parametrizing the normalization curve of D. The number of moduli of W is, by definition the dimension of the image of W with respect to the moduli map. We know that this number is at most equal to seven. In this paper we prove that both irreducible components of S have number of moduli exactly equal to seven.