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Enrico Arbarello

Publications and source records attributed to Enrico Arbarello.

14 recordsLinked to original sources
arXiv2023

A remark on du Val linear systems

Let $|L_g|$, be the genus $g$ du Val linear system on a Halphen surface $Y$ of index $k$. We prove that the Clifford index $cliff(C)$ is constant on smooth curves $C\in |L_g|$. Let $γ(C)$ be the gonality of $C$. When $cliff(C)<\lfloor{\frac{g-1}{2}}\rfloor$ (the relevant case), we show that $γ(C)=cliff(C)+2=k$, and that the gonality is realized by the Weierstrass linear series $|-{kK_Y}_{|C}|$, which is totally ramified at one point. The proof of the first statement follows closely the path indicated by Green and Lazarsfeld for a similar statement regarding K3 surfaces.

math.AG↗
arXiv2023

Singularities of Bridgeland moduli spaces for K3 categories: an update

This survey is a continuation of the study undertaken in \cite{AS18}. We examine the local structure of Bridgeland moduli spaces $M_σ(v,\D)$, where the relevant triangulated category $\D$ is either the bounded derived category $\D=\D^b(X)$ of a K3 surface $X$, or the Kuznetsov component $\D=\Ku(Y)\subset \D ^b(Y)$ of a smooth cubic fourfold $Y\subset \PP^5$. For these moduli spaces, building on \cite{Bmm19}, \cite{Bmm21} we give a direct proof of formality and, using their local isomorphism with quiver varieties, we establish their normality and their irreducibility, as long as $σ$ does not lie on a totally semistable wall. We then connect the variation of GIT quotients for quiver varieties with the changing of stability conditions on moduli spaces.

math.AG↗
arXiv2020

Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach

This paper is withdrawn since we found a flaw in the proof of Theorem 4, asserting that the base locus of the complete linear system of an ample line bundle on a complex abelian variety is reduced. The error is in page 7, line $ -14$, where we claim that the divisor "mathcal E" on the variety $X$ is linearly equivalent to zero. This is untrue. For instance, it would imply that, for a non-torsion point $x$ on an abelian surface $A$, letting $E_x$, $E_{-x}$, and $E_0$ the exceptional curves in the blow up of $A$ at $x$, $-x$, and $0$, then $2E_0$ is linearly equivalent to $E_x +E_{-x}$, which is easily seen to be false. Therefore Theorem 4 of our paper has to be considered unproven. We still believe that it holds true. All the other arguments of our paper are correct but unfortunately they depend on the above mentioned Theorem 4. To be precise, from Theorem 4 follows Theorem 3, asserting that the scheme $\Sigma(X,\Theta, G)$ of Definition 9 is reduced. The rest of the paper contains algebro-geometric proofs of Shiota's theorem characterizing Jacobians via the KP equation (Section 4), and of Krichever's theorems characterizing Jacobians by the existence of an inflectionary or degenerate trisecant to the Kummer variety embedded in $\mathbb P^{2^g-1}$ (Theorem 18 and Theorem 25). Also these proofs are correct but they depend on a weaker version of Theorem 3, namely on the assertion that the components of codimension two of the scheme $\Sigma(X,\Theta, G)$ are generically reduced. In turn, this weaker version of Theorem 3 would follow, by the same argument used in its proof, from a conjecture by Debarre asserting that the base locus of the complete linear system of an ample line bundle on an abelian variety is generically reduced in codimension two.

math.AG↗
arXiv2018

Singularities of moduli spaces of sheaves on K3 surfaces and Nakajima quiver varieties

The aim of this paper is to study the singularities of certain moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non--generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. For sheaves that are pure of dimension one, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety and that, via this isomorphism, the natural symplectic resolutions correspond to variations of GIT quotients of the quiver variety.

math.AG↗
arXiv2016

On hyperplane sections of K3 surfaces

Let C be a Brill-Noether-Petri curve of genus g\geq 12. We prove that C lies on a polarized K3 surface, or on a limit thereof, if and only if the Gauss-Wahl map for C is not surjective. The proof is obtained by studying the validity of two conjectures by J. Wahl. Let I_C be the ideal sheaf of a non-hyperelliptic, genus g, canonical curve. The first conjecture states that, if g\geq 8, and if the Clifford index of C is greater than 2, then H^1(P^{g-1}, I_C^2(k))=0, for k\geq 3. We prove this conjecture for g\geq 11. The second conjecture states that a Brill-Noether-Petri curve of genus g\geq 12 is extendable if and only if C lies on a K3 surface. As observed in the Introduction, the correct version of this conjecture should admit limits of polarised K3 surfaces in its statement. This is what we prove in the present work.

math.AG↗
arXiv2016

Rank two vector bundles on polarised Halphen surfaces and the Gauss-Wahl map for du Val curves

A genus-g du Val curve is a degree-3g plane curve having 8 points of multiplicity g, one point of multiplicity g-1, and no other singularity. We prove that the corank of the Gauss-Wahl map of a general du Val curve of odd genus (>11) is equal to one. This, together with the results of [1], shows that the characterisation of Brill-Noether-Petri curves with non-surjective Gauss-Wahl map as hyperplane sections of K3 surfaces and limits thereof, obtained in [3], is optimal.

math.AG↗
arXiv2016

Explicit Brill-Noether-Petri general curves

Let $p_1,\dots, p_9$ be the points in $\mathbb A^2(\mathbb Q)\subset \mathbb P^2(\mathbb Q)$ with coordinates $$(-2,3),(-1,-4),(2,5),(4,9),(52,375), (5234, 37866),(8, -23), (43, 282), \Bigl(\frac{1}{4}, -\frac{33}{8} \Bigr)$$ respectively. We prove that, for any genus $g$, a plane curve of degree $3g$ having a $g$-tuple point at $p_1,\dots, p_8$, and a $(g-1)$-tuple point at $p_9$, and no other singularities, exists and is a Brill-Noether general curve of genus $g$, while a general curve in that $g$-dimensional linear system is a Brill-Noether-Petri general curve of genus $g$.

math.AG↗
arXiv2014

Mukai's program for curves on a K3 surface

Let C be a general element in the locus of curves in M_g lying on some K3 surface, where g is congruent to 3 mod 4 and greater than or equal to 15. Following Mukai's ideas, we show how to reconstruct the K3 surface as a Fourier-Mukai transform of a Brill-Noether locus of rank two vector bundles on C.

math.AG↗
arXiv2013

Relative Prym varieties associated to the double cover of an Enriques surface

Given an Enriques surface $T$, its universal K3 cover $f: S\to T$, and a genus $g$ linear system $|C|$ on $T$, we construct the relative Prym variety $P_H=\Prym_{v, H}(\D/\CC)$, where $\CC\to |C|$ and $\D\to |f^*C|$ are the universal families, $v$ is the Mukai vector $(0,[D], 2-2g)$ and $H$ is a polarization on $S$. The relative Prym variety is a $(2g-2)$-dimensional possibly singular variety, whose smooth locus is endowed with a hyperkähler structure. This variety is constructed as the closure of the fixed locus of a symplectic birational involution defined on the moduli space $M_{v,H}(S)$. There is a natural Lagrangian fibration $η: P_H \to |C|$, that makes the regular locus of $P_H$ into an integrable system whose general fiber is a $(g-1)$-dimensional (principally polarized) Prym variety, which in most cases is not the Jacobian of a curve. We prove that if $|C|$ is a hyperelliptic linear system, then $P_H$ admits a symplectic resolution which is birational to a hyperkähler manifold of K3$^{[g-1]}$-type, while if $|C|$ is not hyperelliptic, then $P_H$ admits no symplectic resolution. We also prove that any resolution of $P_H$ is simply connected and, when $g$ is odd, any resolution of $P_H$ has $h^{2,0}$-Hodge number equal to one.

math.AG↗
arXiv2012

Two remarks on the Weierstrass flag

We show that the locally closed strata of the Weierstrass flags on the moduli spaces of curves of genus g and on the moduli space of curves of genus g with one marked point are almost never affine.

math.AG↗
arXiv2009

Teichmueller space via Kuranishi families

We construct Teichmueller space by patching together Kuranishi families. We also discuss the basic properties of Teichmueller space, and in particular show that our construction leads to simplifications in the proof of Teichmueller's theorem asserting that the genus g Teichmueller space is homeomorphic to a (6g-6)-dimensional ball.

math.AG↗
arXiv2008

Divisors in the moduli spaces of curves

In this mostly expository paper we review several known results about the cohomology of moduli spaces of smooth and stable curves, focusing in particular on low degree cohomology. We also give a new proof of Harer's theorem describing the second cohomology group of the moduli space of smooth n-pointed curves of given genus

math.AG↗
arXiv1994

Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves

Based on the combinatorial description of the moduli spaces of curves provided by Strebel differentials, Witten and Kontsevich have introduced combinatorial cohomology classes $W_{(m_0,m_1,m_2,\dots),n}$, and conjectured that these can be expressed in terms of Mumford-Morita-Miller classes. It is argued that this link should be provided by a theorem of Di Francesco, Itzykson and Zuber which relates the derivatives of the Witten-Kontsevich partition function with respect to one set of variables to the derivatives with respect to the other set of variables. Two things are shown. First of all that this works in complex codimension 1. Secondly that in all the cases when it has been possible to make the Di Francesco, Itzykson and Zuber correpondence explicit this translates into identities of the type $$ \int_{W_{(m_0,m_1,m_2,\dots),n}}\prodψ_i^{d_i} =\int_{\overline{\cal{M}}_{g,n}} X_{(m_0,m_1,m_2,\dots),n}\prodψ_i^{d_i} $$ where the $X_{(m_0,m_1,m_2,\dots),n}$ are explicit polynomials in the algebro-geometric classes and the $ψ_i$ are the Chern classes of the point bundles, for any choice of $d_1,\dots,d_n$.

alg-geom↗