Let $Y$ be an integral projective curve with $g:=p_a\left(Y\right)\ge 2$. For all positive integers $d$, $r$ let $W_d^r\left(Y\right)\left({}^{*A*}\right)$ be the set of all $L\in {\text{Pic}}^d\left(Y\right)$ with $h^0\left(Y,L\right)\ge r+1$ and $L$ spanned. Here we prove that if $d\le g-2$, then $dim\left(W_d^r\left(Y\right)\left({}^{*A*}\right)\right)\le d-3r$ except in a few cases (essentially if $Y$ is a double covering).

Sia $X\subset \mathbb{P}\left(H^0{\left(X,L\right)}^{*}\right)$ una curva proeittiva e lissa, generali nel senso di Brill-Noether, indichiamo con ${\mathcal{R}}_4\left(X\right)$ l'insieme algebrico di quadrici di rango $4$ contenendo a $X$. In questo lavoro noi descriviamo birazionalmente i componenti irriducibile di ${\mathcal{R}}_4\left(X\right)$.

Let $Y$ be a normal crossing divisor in the smooth complex projective algebraic variety $X$ and let $U$ be a tubular neighbourhood of $Y$ in $X$. Using geometrical properties of different intersections of the irreducible components of $Y$, and of the embedding $Y\subset X$, we provide the “normal forms” of a set of geometrical cycles which generate $H_{*}(A,B)$, where $(A,B)$ is one of the following pairs $(Y,\varnothing )$, $(X,Y)$, $(X,X-Y)$, $(X-Y,\varnothing )$ and $(\partial U,\varnothing )$. The construction is compatible with the weights in $H_{*}(A,B,\mathbb{Q})$ of Deligne’s mixed Hodge structure. The main technical part...

This article gives a description, by means of functorial intrinsic fibrations, of the geometric structure (and conjecturally also of the Kobayashi pseudometric, as well as of the arithmetic in the projective case) of compact Kähler manifolds. We first define special manifolds as being the compact Kähler manifolds with no meromorphic map onto an orbifold of general type, the orbifold structure on the base being given by the divisor of multiple fibres. We next show that rationally connected Kähler...

For any compact Kähler manifold $X$ and for any equivalence relation generated by a symmetric binary relation with compact analytic graph in $X\times X$, the existence of a meromorphic quotient is known from Inv. Math. 63 (1981). We give here a simplified and detailed proof of the existence of such quotients, following the approach of that paper. These quotients are used in one of the two constructions of the core of $X$ given in the previous paper of this fascicule, as well as in many other questions.