Curves in the double plane

We study in detail locally Cohen-Macaulay curves in P⁴ which are contained in a double plane 2H, thus completing the classification of curves lying on surfaces of degree two. We describe the irreducible components of the Hilbert schemes H _d,g(2H) of lo-cally Cohen-Macaulay curves in 2H of degree d and arithmetic genus g, and we show that H _d,g(2H) is connected. We also discuss the Rao module of these curves and liaison and biliaison equiva-lence classes.     

On Some Moduli Spaces of Bundles on K3 Surfaces

We give infinitely many examples in which the moduli space of rank 2 H-stable sheaves on a K3 surface S endowed by a polarization H of degree 2g – 2, with Chern classes c₁ = H and c₂ = g – 1, is birationally equivalent to the Hilbert scheme S[g – 4] of zero dimensional subschemes of S of length g – 4. We get in this way a partial generalization of results from [5] and [1].     

Imposing Singular Points and Many Nodes to Curves on Surfaces

 Let S be a smooth projective surface. Here we study the conditions imposed to curves of a fixed very ample linear system by a general union of types of singularities τ when most of connected components of τ are ordinary double points. This problem is related to the existence of “good” families of curves on S with prescribed singularities, most of them being nodes, and to the regularity of their Hilbert scheme. Received 6 July 2000; in revised form 16 June 2001     

Hilbert Scheme of a Pair of Codimension Two Linear Subspaces

We study the component H _n of the Hilbert scheme whose general point parameterizes a pair of codimension two linear subspaces in ? ⁿ for n ≥ 3. We show that H _n is smooth and isomorphic to the blow-up of the symmetric square of 𝔾(n ? 2, n) along the diagonal. Further H _n intersects only one other component in the full Hilbert scheme, transversely. We determine the stable base locus decomposition of its effective cone and give modular interpretations of the corresponding models, hence conclude that H _n is a Mori dream space.     

Stratification of the Hilbert scheme of space curves with a stable normal bundle

Ford≥3g and 1≤s≤[g/2], we study the strataN _{d, g(s)} of degreed genusg spaces curvesC whose normal bundleN _C is stable with stability degree (integer of Lange-Narasimhan) σ(N _C)=2s. We prove thatN _{d, g(s)} has an irreducible component of the right dimension whose general curve has a normal bundle with the right number of maximal subbundles. We consider also the semi-stable case (s=0), obtaining similar results. We prove our results by studying the normal bundles of reducible curves and their deformations. Both authors were partially supported by MIUR and GNSAGA of INdAM (Italy).     

A characterization of bielliptic curves and applications to their moduli spaces

We deal with the covers of degree 4 naturally associated to a bielliptic curve of genus g≥6, giving a proof of the unirationality of the moduli space ? _g ^be of such curves, of the rationality of the Hurwitz scheme ℌ ^be _{4, g} of bielliptic curves of even genus g, whereas, when g is odd, we construct a finite map ℂ^{2 g -2}→? _g ^be and compute its degree. Received: March 25, 2000; in final form: March 10, 2001?Published online: May 29, 2002     

Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">d</Emphasis></Subscript>

Let Y be a smooth, connected, projective complex curve. In this paper, we study the Hurwitz spaces which parameterize branched coverings of Y whose monodromy group is a Weyl group of type D _d and whose local monodromies are all reflections except one. We prove the irreducibility of these spaces when and successively we extend the result to curves of genus g ≥ 1.     

Non-special scrolls with general moduli

Abstract  In this paper we study smooth, non-special scrolls S of degree d, genus g ≥ 0, with general moduli. In particular, we study the scheme of unisecant curves of a given degree on S. Our approach is mostly based on degeneration techniques. Keywords Ruled surfaces, Hilbert schemes of scrolls, Moduli, Embedded degenerations Mathematics Subject Classification (2000) 14J26, 14D06, 14C20, (Secondary) 14H60, 14N10     

The cup product of Hilbert schemes for K3 surfaces

To any graded Frobenius algebra A we associate a sequence of graded Frobenius algebras A ^[n] so that there is canonical isomorphism of rings (H ^*(X;ℚ)[2]) ^[n] ≅H ^*(X ^[n] ;ℚ)[2n] for the Hilbert scheme X ^[n] of generalised n-tuples of any smooth projective surface X with numerically trivial canonical bundle. Oblatum 25-I-2001 & 18-IX-2002?Published online: 24 February 2003     

Hilbert Schemes of Degree Four Curves

In this paper we determine the irreducible components of the Hilbert schemes H _4,g of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g: there are roughly (g ²/24) of them, most of which are families of multiplicity structures on lines. We give deformations which show that these Hilbert schemes are connected. For g–3 we exhibit a component that is disjoint from the component of extremal curves and use this to give a counterexample to a conjecture of Aït-Amrane and Perrin.     

Obstructions to deforming curves on a prime Fano 3‐fold

We prove that for every smooth prime Fano 3‐fold V, the Hilbert scheme of smooth connected curves on V contains a generically non‐reduced irreducible component of Mumford type. We also study the deformations of degenerate curves C in V, i.e., curves C contained in a smooth anticanonical member of V. We give a sufficient condition for C to be stably degenerate, i.e., every small (and global) deformation of C in V is contained in a deformation of S in V. As a result, by using the Hilbert‐flag scheme of V, we determine the dimension and the smoothness of at the point [C], assuming that the class of C in is generated by together with the class of a line, or a conic on V.     

On generically non‐reduced components of Hilbert schemes of smooth curves

A classical example of Mumford gives a generically non‐reduced component of the Hilbert scheme of smooth curves in such that a general element of the component is contained in a smooth cubic surface in . In this article we use techniques from Hodge theory to give further examples of such (generically non‐reduced) components of Hilbert schemes of smooth curves without any restriction on the degree of the surface containing it. As a byproduct we also obtain generically non‐reduced components of certain Hodge loci.     

A stratification of Hilbert schemes by initial ideals and applications

We show that the set of the homogeneous saturated ideals with given initial ideal in a fixed term-ordering is locally closed in the Hilbert scheme, and that it is affine if the initial ideal is saturated. Then, Hilbert schemes can be stratified using these subschemes. We investigate the behaviour of this stratification with respect to some properties of the closed points. As application, we describe the singular locus of the component of Hilb^{4 z +1} _ℙ4 containing the ACM curves of degree 4. Received: 30 November 1998 / Revised version: 16 September 1999     

Compactifying moduli of hyperelliptic curves

We construct a new compactification of the moduli spaceH _g of smooth hyperelliptic curves of genusg. We compare our compactification with other well-known remarkable compactifications ofH _g. The author was partially supported byCNP _q, Proc. 151610/2005-3, and by Faperj, Proc. E-26/152-629/2005.     

On Some Moduli Spaces of Bundles on K3 Surfaces

We give infinitely many examples in which the moduli space of rank 2 H-stable sheaves on a K3 surface S endowed by a polarization H of degree 2g – 2, with Chern classes c₁ = H and c₂ = g – 1, is birationally equivalent to the Hilbert scheme S[g – 4] of zero dimensional subschemes of S of length g – 4. We get in this way a partial generalization of results from [5] and [1].     

Remarks on families of singular curves with hyperelliptic normalizations

We give restrictions on the existence of families of curves on smooth projective surfaces S of nonnegative Kodaira dimension all having constant geometric genus p_g ? 2 and hyperelliptic normalizations. In particular, we prove a Reider-like result that relies on deformation theory and bending-and-breaking of rational curves in Sym²(S). We also give examples of families of such curves.     

On the Lüroth semigroup of curves lying on a smooth quadric

For a smooth irreducible complete algebraic curveC the “gaps” are the integersn such that every linear series of degreen has at least a base point. The Lüroth semigroup S_C of a curveC is the subsemigroup ofN whose elements are not gaps. In this paper we deal with irreducible smooth curves of type (a, b) on a smooth quadricQ. The main result is an algorithm by which we can say if some integer λ∈N is a gap or is in S_C. In the general case there are integers λ which are undecidable. For curves such as complete intersection, arithmetically Cohen-Macaulay or Buchsbaum, we are able to describe explicitly “intervals” of gaps and “intervals” of integers which belong to S_C. For particular cases we can completely determine S_C, by giving just the type of the curve (in particular the degree and the genus). Work done with financial support of M.U.R.S.T. while the authors were members of G.N.S.A.G.A. of C.N.R.     

Irreducibility of the Hilbert scheme of smooth curves in $$\mathbb {P}^3$$ of de gree g and genus g

We denote by \(\mathcal {H}_{d,g,r}\) the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree d and genus g in \(\mathbb {P}^r\). In this note, we show that any non-empty \(\mathcal {H}_{g,g,3}\) is irreducible without any restriction on the genus g. This extends the result obtained earlier by Iliev (Proc Am Math Soc 134:2823–2832, 2006).     

Imposing Singular Points and Many Nodes to Curves on Surfaces

 Let S be a smooth projective surface. Here we study the conditions imposed to curves of a fixed very ample linear system by a general union of types of singularities τ when most of connected components of τ are ordinary double points. This problem is related to the existence of “good” families of curves on S with prescribed singularities, most of them being nodes, and to the regularity of their Hilbert scheme.