Abstract. We treat the problem of completing the moduli space for roots of line bundles on curves. Special attention is devoted to higher spin curves within the universal Picard scheme. Two new different constructions, both using line bundles on nodal curves as boundary points, are carried out and compared with pre-existing ones.

Résumé. Généralisant une question de Mukai, nous conjecturons qu'une variété de Fano X de nombre de Picard ρ X et de pseudo-indice ι X vérifie ρ X (ι X − 1) ≤ dim(X). Nous démontrons cette conjecture dans plusieurs situations : X est une variété de Fano de dimension ≤ 4, X est une variété de Fano torique de dimension ≤ 7 ou X est une variété de Fano torique de dimension arbitraire avec ι X ≥ dim(X)/3 + 1. Enfin, nous présentons une approche nouvelle pour le cas général.Abstract. Generalizing a question of Mukai, we conjecture that a Fano manifold X with Picard number ρ X and pseudo-index ι X satisfies ρ X (ι X − 1) ≤ dim(X). We prove this inequality in several situations : X is a Fano manifold of dimension ≤ 4, X is a toric Fano manifold of dimension ≤ 7 or X is a toric Fano manifold of arbitrary dimension with ι X

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In this paper we describe the geometry of the 2m-dimensional Fano manifold G parametrizing (m − 1)-planes in a smooth complete intersection Z of two quadric hypersurfaces in the complex projective space P 2m+2 , for m ≥ 1. We show that there are exactly 2 2m+2 distinct isomorphisms in codimension one between G and the blow-up of P 2m at 2m + 3 general points, parametrized by the 2 2m+2 distinct m-planes contained in Z, and describe these rational maps explicitly. We also describe the cones of nef, movable and effective divisors of G, as well as their dual cones of curves. Finally, we determine the automorphism group of G.These results generalize to arbitrary even dimension the classical description of quartic del Pezzo surfaces (m = 1).

Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in X. We consider the image N 1 (D, X) of N 1 (D) in N 1 (X) under the natural pushforward of 1-cycles. We show that ρwhere S is a Del Pezzo surface, or codim N 1 (D, X) = 3 and X has a fibration in Del Pezzo surfaces onto a Fano manifold T such that ρ X − ρ T = 4. RésuméSoit X une variété de Fano lisse et complexe de dimension arbitraire, et D un diviseur premier dans X. Nous considérons l'image N 1 (D, X) de N 1 (D) dans N 1 (X) par l'application naturelle de "push-forward" de 1-cycles. Nous démontrons que ρ X − ρ D ≤ codim N 1 (D, X) ≤ 8. De plus, si codim N 1 (D, X) ≥ 3, alors soit X ∼ = S × T où S est une surface de Del Pezzo, soit codim N 1 (D, X) = 3 et X a une fibration en surfaces de Del Pezzo sur une variété de Fano lisse T , telle que ρ X − ρ T = 4.

Abstract. Given a covering family V of effective 1-cycles on a complex projective variety X, we find conditions allowing one to construct a geometric quotient q : X → Y , with q regular on the whole of X, such that every fiber of q is an equivalence class for the equivalence relation naturally defined by V . Among other results, we show that on a normal and Q-factorial projective variety X with canonical singularities and dim X ≤ 4, every covering and quasi-unsplit family V of rational curves generates a geometric extremal ray of the Mori cone NE(X) of classes of effective 1-cycles and that the associated Mori contraction yields a geometric quotient for V .

Let X be a smooth, complete toric variety. Let A 1 (X) be the group of algebraic 1-cycles on X modulo numerical equivalence and N 1 (X) = A 1 (X) ⊗ Z Q . Consider in N 1 (X) the cone NE(X) generated by classes of curves on X. It is a well-known result due to M. Reid [13] that NE(X) is closed, polyhedral and generated by classes of invariant curves on X. The variety X is projective if and only if NE(X) is strictly convex; in this case, a 1-dimensional face of NE(X) is called an extremal ray. It is shown in [13] that every extremal ray admits a contraction to a projective toric variety.We think of A 1 (X) as a lattice in the Q -vector space N 1 (X). Suppose that X is projective. For every extremal ray R ⊂ NE(X), we choose the primitive class in R ∩ A 1 (X); we call this class an extremal class. The set E of extremal classes is a generating set for the cone NE(X), namely NE(X) = γ∈E Q ≥0 γ. For many purposes it would be useful to have a linear decomposition with integral coefficients: for instance, what can we say about curves having minimal degree with respect to some ample line bundle on X? It is an open question whether extremal classes generate NE(X) ∩ A 1 (X) as a semigroup. In this paper we introduce a set C ⊇ E of classes in NE(X) ∩ A 1 (X) which is a set of generators of NE(X) ∩ A 1 (X) as a semigroup. Classes in C are geometrically characterized by "contractibility": Definition 2.3. Let γ ∈ NE(X)∩A 1 (X) be primitive along A 1 (X)∩Q ≥0 γ and such that there exists some irreducible curve in X having numerical class in Q ≥0 γ. We say that γ is contractible if there exist a toric variety X γ and an equivariant morphism ϕ γ : X → X γ , with connected fibers, such that forThis definition does not need the projectivity of X. We give a combinatorial characterization of contractibility in terms of the fan of X, and we show that a class γ is contractible if and only if every irreducible invariant curve in the class is extremal in every irreducible invariant surface containing it (Theorem 2.2). In the projective case, this property is false for extremal classes: it can happen that every invariant curve in a class is extremal in every invariant subvariety containing it, but the class is not extremal in X (see example on page 106).When X is projective, all extremal classes are contractible, and a contractible class γ is extremal if and only if X γ is projective. Hence, contractible non-extremal classes correspond to birational contractions to nonprojective toric varieties (Corollary 3.3). Moreover, we show that a class γ ∈ NE(X) is contractible if and only if it is extremal in the subvariety A given by the intersection of all irreducible invariant divisors having negative intersection with γ.As mentioned above, the main result of the paper is that when X is projective, contractible classes span A 1 (X) ∩ NE(X) as a semigroup (Theorem 4.1), namely every class in A 1 (X) ∩ NE(X) decomposes as a linear combination with positive integral coefficients of contractible classes. In the non-projective case, the situation is ve...

The uv induced polymerization of monolayers of n-octadecyl methacrylate and l-n-octadecyloxy-2,3-diacryloyloxypropane at water-gas interface has been studied. In each case, the compressibility and the surface viscosity of the film at different stages of irradiation were measured. The main difference between monoand diacrylic esters is the occurrence of a divergence of the viscosity coefficient during the polymerization process for the second compound. In the light of our experimental results, we discuss the mechanism of the photochemical reaction, the role played by chain entanglements, and the specific effects of cross linking in the diacrylic ester.