Postulation of subschemes of irreducible curves on a quadric surface

We study all the possible Hilbert functions of 0-dimensional subschemes of irreducible curves of a smooth quadric of ?³. We obtain characterizations in case of complete intersection, arithmetically Cohen-Macaulay and arithmetically Buchsbaum curves and other necessary conditions in the general cases.     

Postulation of subschemes of irreducible curves on a quadric surface

We study all the possible Hilbert functions of 0-dimensional subschemes of irreducible curves of a smooth quadric of ℙ³. We obtain characterizations in case of complete intersection, arithmetically Cohen-Macaulay and arithmetically Buchsbaum curves and other necessary conditions in the general cases.     

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.     

Curve aritmeticamente Buchsbaum su superfici di grado 3 e 4 dello spazio proiettivo

Riassunto Si studiano curve aritmeticamente Buchsbaum nello spazio proiettivoP ³, tali che l’ordine minimo di una superficie che le contiene è 3 o 4. Per tali curve si determinano l’ordine, il genere aritmetico, il carattere numerico connesso, il modulo di Hartshorne-Rao e la curva legata di ordine minimo. Nel caso di curve situate su superfici cubiche lisce si determinano anche i multigradi corrispondenti.

---

Summary In this paper we study arithmetically Buchsbaum curves in the projective spaceP ³, such that the minimal degree of a surface containing them is 3 or 4. For such curves we determine the degree, the aritmethic genus, the connected numerical character, the Hartshorne-Rao module, and the linked curves having minimal degree. For curves lying on smooth cubic surfaces ofP ³ we determine also the associated multidegrees.

---

---

Lavoro eseguito sotto gli auspici del G.N.S.A.G.A. del C.N.R.     

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.     

Surfaces with pg = 2, K2 = 3 and a pencil of curves of genus 2

We classify minimal smooth surfaces of general type with K ² = 3, p _g = 2 which admit a fibration of curves of genus 2.We prove that they form an irreducible set of dimension 22 in their moduli space.      

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.     

Rational curves of degree at most 9 on a general quintic threefold

ABSTRACT.

We prove the following form of the Clemens conjecture in low degree. Let d ≤ 9, and let F be a general quintic threefold in P ⁴. Then (1) the Hilbert scheme of rational, smooth and irreducible curves of degree d on F is finite, nonempty, and reduced; moreover, each curve is embedded in F with normal bundle (?1) ⊕ (?1), and in P ⁴ with maximal rank. (2) On F, there are no rational, singular, reduced and irreducible curves of degree d, except for the 17,601,000 six-nodal plane quintics (found by Vainsencher). (3) On F, there are no connected, reduced and reducible curves of degree d with rational components.     

On the Hilbert scheme of curves in higher-dimensional projective space

In this paper we prove that, for anyn≥3, there exist infinitely manyr∈N and for each of them a smooth, connected curveC _r in ℙ ^r such thatC _r lies on exactlyn irreducible components of the Hilbert scheme Hilb(ℙ ^r ). This is proven by reducing the problem to an analogous statement for the moduli of surfaces of general type.     

Smooth approximation of mather sets of monotone twist mappings

In the theory of monotone twist mappings of a cylinder one constructs for every rotation number α invariant minimal sets M_α. In this paper an approximation of these Mather sets by smooth invariant curves M^v_α is devised, which for v → 0 converge to M_α almost everywhere. The main point of the construction is that the approximating curves M^v_α form for v > 0 a smooth foliation. The approximation is achieved with the help of a regularized version of the Percival variational problem. © 1994 John Wiley & Sons, Inc.     

On Buchsbaum bundles on quadric hypersurfaces

Let E be an indecomposable rank two vector bundle on the projective space ℙ ⁿ , n ≥ 3, over an algebraically closed field of characteristic zero. It is well known that E is arithmetically Buchsbaum if and only if n = 3 and E is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface Q _n ⊂ ℙ ⁿ⁺¹, n ≥ 3. We give in fact a full classification and prove that n must be at most 5. As to k-Buchsbaum rank two vector bundles on Q ₃, k ≥ 2, we prove two boundedness results.     

Fragmented deformations of primitive multiple curves

A primitive multiple curve is a Cohen-Macaulay irreducible projective curve Y that can be locally embedded in a smooth surface, and such that Y _red is smooth. We study the deformations of Y to curves with smooth irreducible components, when the number of components is maximal (it is then the multiplicity n of Y). We are particularly interested in deformations to n disjoint smooth irreducible components, which are called fragmented deformations. We describe them completely. We give also a characterization of primitive multiple curves having a fragmented deformation.     

Some remarks on quasi-complete intersection space curves

Summary This paper is devoted to the study of quasi-complete intersection space curves. First, we give a Castelnuovo bound on the index of regularity fork-Buchsbaum, quasi-complete intersection space curves. Then, we prove that, smooth, arithmetically Buchsbaum, quasi-complete intersection space curves of maximal rank are unobstructed. We conclude by studying some examples and adding some remarks.

---

Riassunto Questo articolo è dedicato allo studio delle curve spaziali quasi-complete intersezioni. Dapprima, noi diamo un limite di Castelnuovo per l'indice di regolarità delle curvek-Buchsbaum, quasi-complete intersezioni. Inoltre, dimostriamo che le curve liscie, aritmeticamente di Buchsbaum, di rango massimo quasi complete intersezioni sono non ostruite. Si conclude studiando alcuni esempi e aggiungendo alcune osservazioni.

---

---

To Joan     

Reducibility of Punctual Hilbert Schemes of Cone Varieties

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ? ? ^N of dimension n and degree d and an integer s ₀ such that Hilb ^s (X) is reducible for all s ≥ s ₀. X will be a projective cone in ? ^N over an arbitrary projective variety Y ? ? ^N?1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points.     

The Dimension of the Hilbert Scheme of Special Threefolds #

The Hilbert scheme of 3-folds in ? ⁿ , n ≥  6 , that are scrolls over ? ² or over a smooth quadric surface Q  ? ? ³ or that are quadric or cubic fibrations over ? ¹ is studied. All known such threefolds of degree 7  ≤ d ≤  11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed.     

Linear series on a general k-gonal curve

This paper is a contribution towards a Brill-Noether theory for the moduli space of smooth &-gonal curves of genusg. Specifically, we prove the existence of certain special divisors on a generalk-gonal curveC of genusg, and we detect an irreducible component of the “expected” dimension in the varietyW ^r _d (C), (r ≤k — 2) of special divisors ofC. The latter induces a new proof of the existence theorem for special divisors on a smooth curve.     

Self-dual representations of some dyadic groups

Let F be a non-Archimedean local field of residual characteristic two and let d be an odd positive integer. Let D be a central F-division algebra of dimension d ². Let π be one of: an irreducible smooth representation of D  ^× , an irreducible cuspidal representation of GL _d (F), an irreducible smooth representation of the Weil group of F of dimension d. We show that, in all these cases, if π is self-contragredient then it is defined over \mathbb Q{\mathbb Q} and is orthogonal. We also show that such representations exist.     

Non-constant curves of genus 2 with infinite pro-Galois covers

For every odd prime number p, we give examples of non-constant smooth families of curves of genus 2 over fields of characteristic p which have pro-Galois (pro-étale) covers of infinite degree with geometrically connected fibers. The Jacobians of the curves are isomorphic to products of elliptic curves.     

Globally Irreducible Representations of Finite Groups and Integral Lattices

The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell--Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L ₂(q) and ²B₂(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (pⁿ-1)/2 of the symplectic group Sp_2n(p) (p 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pⁿ–1). A summary of currently known globally irreducible representations is given.     

Isomorphism rigidity of irreducible algebraic ℤd-actions

An irreducible algebraic ℤ ^d -actionα on a compact abelian group X is a ℤ ^d -action by automorphisms of X such that every closed, α-invariant subgroup Y⊊X is finite. We prove the following result: if d≥2, then every measurable conjugacy between irreducible and mixing algebraic ℤ ^d -actions on compact zero-dimensional abelian groups is affine. For irreducible, expansive and mixing algebraic ℤ ^d -actions on compact connected abelian groups the analogous statement follows essentially from a result by Katok and Spatzier on invariant measures of such actions (cf. [4] and [3]). By combining these two theorems one obtains isomorphism rigidity of all irreducible, expansive and mixing algebraic ℤ ^d -actions with d≥2. Oblatum 30-IX-1999 & 4-V-2000?Published online: 16 August 2000