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Marco Franciosi

Publications and source records attributed to Marco Franciosi.

At least 19 recordsLinked to original sources
arXiv2026

Mok tensors and Orbifold Quotients of Bounded symmetric domains without ball factors

In this paper we characterize the compact orbifolds, quotients $ X = \mathcal{D}/ Γ$ of a bounded symmetric domain $\mathcal{D}$ with no higher dimensional ball factor by the action of a discontinuous group $Γ$, as those projective orbifolds with ample orbifold canonical divisor which admit a Mok curvature type tensor of orbifold type and satisfying certain other conditions implying the existence of a finite smooth covering.

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arXiv2024

2-Gorenstein stable surfaces with $K_X^2 = 1$ and $χ(X) = 3$

The compactification $\overline M_{1,3}$ of the Gieseker moduli space of surfaces of general type with $K_X^2 =1 $ and $χ(X)=3$ in the moduli space of stable surfaces parametrises so-called stable I-surfaces. We classify all such surfaces which are 2-Gorenstein into four types using a mix of algebraic and geometric techniques. We find a new divisor in the closure of the Gieseker component and a new irreducible component of the moduli space.

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arXiv2022

Degeneration of Hodge structures on I-surfaces

Work of Green, Griffiths, Laza, and Robles suggests that the moduli space of (smoothable) stable surfaces should admit a natural stratification defined via Hodge theoretic data. In the case of stable surfaces with $K_X^2 = 1$ and $χ(X) = 3$ we compute the Hodge type of all examples known to us and show that all predicted degenerations are geometrically realised.

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arXiv2021

On T-divisors and intersections in $\overline{M}_{1,3}$

The moduli space of stable surfaces with $K_X^2 = 1$ and $χ(X) = 3$ has at least two irreducible components that contain surfaces with T-singularities. We show that the two known components intersect transversally in a divisor. Moreover, we exhibit other new boundary divisors and study how they intersect one another.

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arXiv2021

Deformations of semi-smooth varieties

For a singular variety X, an essential step to determine its smoothability and study its deformations is the understanding of the tangent sheaf and of the sheaf T^1_X:=ext^1(Omega_X,O_X). A variety is semi-smooth if its singularities are étale locally the product of a double crossing point (uv=0) or a pinch point (u^2-v^2w=0) with affine space; equivalently, if it can be obtained by gluing a smooth variety along a smooth divisor via an involution with smooth quotient. Our main result is the explicit computation of the tangent sheaf and the sheaf T^1_X for a semi-smooth variety X in terms of the gluing data.

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arXiv2021

Smoothing semi-smooth Stable Godeaux surfaces

We show that all the semi-smooth stable complex Godeaux surfaces, classified in [FPR18a], are smoothable, and that the moduli stack is smooth of the expected dimension 8 at the corresponding points.

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arXiv2020

I-surfaces with one T-singularity

We classify normal stable surfaces with $K_X^2 = 1$, $p_g = 2$ and $q=0$ with a unique singular point which is a non-canonical T-singularity, thus exhibiting two divisors in the main component and a new irreducible component of the moduli space of stable surfaces $\overline\gothM_{1,3}$.

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arXiv2018

Scrolls containing binary curves

We study families of scrolls containing a given rational curve and families of rational curves contained in a fixed scroll via a stratification in terms of the degree of the induced map onto P^1 and we prove that there is no rational normal scroll of minimal degree and of dimension <=n/2 containing a general binary curve inP^n.

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arXiv2018

Clifford index for reduced curves

We extend the notion of Clifford index to reduced curves with planar singularities by considering rank 1 torsion free sheaves. We investigate the behaviour of the Clifford index with respect to the combinatorial properties of the curve and we show that Green's conjecture holds for certain classes of curves given by the union of two irreducible components.

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arXiv2016

Gorenstein stable Godeaux surfaces

We classify Gorenstein stable numerical Godeaux surfaces with worse than canonical singularities and compute their fundamental groups.

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arXiv2016

$(p,d)$-elliptic curves of genus two

We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree $p$, resp. $d$, onto smooth elliptic curves, with particular attention to the case $p$ prime.

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arXiv2015

Gorenstein stable surfaces with $K_X^2 = 1$ and $p_g>0$

In this paper we consider Gorenstein stable surfaces with $K^2_X=1$ and positive geometric genus. Extending classical results, we show that such surfaces admit a simple description as weighted complete intersection. We exhibit a wealth of surfaces of all possible Kodaira dimensions that occur as normalisations of Gorenstein stable surfaces with $K_X^2=1$; for $p_g=2$ this leads to a rough stratification of the moduli space. Explicit non-Gorenstein examples show that we need further techniques to understand all possible degenerations.

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arXiv2014

Green's conjecture for binary curves

We show that Generic Green's conjecture holds for generic binary curves, through a detailed analysis of the family of scrolls containing fixed rational normal curves.

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arXiv2014

Computing invariants of semi-log-canonical surfaces

We describe some methods to compute fundamental groups, (co)homology, and irregularity of semi-log-canonical surfaces. As an application, we show that there are exactly two irregular Gorenstein stable surfaces with $K^2=1$, both of which have $χ(X) = 0$ and $\mathrm{Pic}^0(X)=\mathbb{C}^*$ but different homotopy type.

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arXiv2013

The canonical ring of a 3-connected curve

Let C be a projective curve either reduced with planar singularities or contained in a smooth algebraic surface. We show that the canonical ring R(C, ω_C)= \oplus_{k \geq 0} H^0(C, ω_C^k is generated in degree 1 if C is 3-connected and not (honestly) hyperelliptic; we show moreover that R(C, L)=\oplus_{k \geq 0} H^0(C,L^k)$ is generated in degree 1 if C is reduced with planar singularities and L is an invertible sheaf such that deg L_{|B} \geq 2p_a(B)+1 for every B \subseteq C.

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