We prove the existence of a smooth and non-degenerate curve $X\subset \mathbb{P}^{n}$, $n\geqslant 8$, with $\deg (X)=d$, $p_{a}(X)=g$, $h^{1}(N_{X}(-1))=0$, and general moduli for all $(d,g,n)$ such that $d\geqslant (n-3)\lceil g/2\rceil +n+3$. It was proved by C. Walter that, for $n\geqslant 4$, the inequality $2d\geqslant (n-3)g+4$ is a necessary condition for the existence of a curve with $h^{1}(N_{X}(-1))=0$.

We prove that a reduced and irreducible algebraic surface in $\mathbb{CP}^{3}$ containing infinitely many twistor lines cannot have odd degree. Then, exploiting the theory of quaternionic slice regularity and the normalisation map of a surface, we give constructive existence results for even degrees.

We give a complete classification of globally generated vector bundles of rank 3 on a smooth quadric threefold with c 1 ≤ 2 and extend the result to arbitrary higher rank case. We also investigate the existence of globally generated indecomposable vector bundles, and give the sufficient and necessary conditions on numeric data of vector bundles for indecomposability.

Let V be an infinite-dimensional locally convex complex space, X a closed subset of P(V) defined by finitely many continuous homogeneous equations and E a holomorphic vector bundle on X with finite rank. Here we show that E is holomorphically trivial if it is topologically trivial and spanned by its global sections and in a few other cases.

Let $X,\,Y$ be reduced and irreducible compact complex spaces and $S$ the set of all isomorphism classes of reduced and irreducible compact complex spaces $W$ such that $X\,\times \,Y\,\cong \,X\,\times \,W$ . Here we prove that $S$ is at most countable. We apply this result to show that for every reduced and irreducible compact complex space $X$ the set $S(X)$ of all complex reduced compact complex spaces $W$ with $X\,\times \,{{X}^{\sigma }}\,\cong \,W\,\times \,{{W}^{\sigma }}$ (where ${{A}^{\sigma }}$ denotes the complex conjugate of any variety $A$ ) is at most countable.

Here we study the dimension δ(m, X) of the general fibers of the m-Gaussian map of a singular n-dimensional variety X ⊂ Pn. We show that for all integers a, b, c, d with n ≦ a < b ≦ c < d ≦ N − 1 and a + d = b + c we have δ (a, X) + δ(d, X) > δ(b, X) + δ(c, X). If δ(X, N − 1) is very large we give some classification results which extend to the singular case some results of Ein.

Let X be a smooth complex projective curve of genus g [ges ] 1. If g [ges ] 2, then assume further that X is either bielliptic or with general moduli. Fix integers r, s, a, b with r > 1, s > 1 and as [les ] br. Here we prove the existence of an exact sequence

formula here

of semistable vector bundles on X with rk(H) = r, rk(Q) = s, deg(H) = a and deg(Q) = b.

Let $E$ be a stable rank 2 vector bundle on a smooth projective curve $X$ and $V\,\left( E \right)$ be the set of all rank 1 subbundles of $E$ with maximal degree. Here we study the varieties (non-emptyness, irreducibility and dimension) of all rank 2 stable vector bundles, $E$ , on $X$ with fixed $\deg \left( E \right)$ and $\deg \left( L \right),\,L\,\in \,V\left( E \right)$ and such that $\text{card}\,\left( V(E) \right)\,\ge \,2\,(\text{resp}\text{. card}\left( V(E) \right)\,=\,2)$ .

Let X be a smooth projective curve of genus g ≥ 2. Here we construct stable vector bundles on X equipped with a filtration with suitable numerical properties and with stable graded subquotients.

In this paper we will study the Brill–Noether theory of vector bundles on a smooth projective curve X. As usual in papers on this topic we are mainly interested in stable or at least semistable bundles. Let Wkr, d(X) be the scheme of all stable vector bundles E on X with rank (E)=r, deg (E)=d and h0(X, E)[ges ]k+1. For a survey of the main known results, see the introduction of [6]. The referee has pointed out that the results in [6] were improved by V. Mercat in [14]; he proved that Wkr, d(X) is non-empty for d<2r if and only if k+1[les ]r+(d−r)/g. If X has general moduli the more interesting existence theorem was proved in [19]. However, in this paper we are mainly interested in very special curves X, e.g. the hyperelliptic or the bielliptic curves. We work over an algebraically closed base field K. In Section 5 we will assume char (K)=0. In Section 1 we will give some theorems of Clifford's type. In Section 2 we will construct several stable bundles with certain properties. Here the main tool is an operation (the +elementary transformation) which sends a vector bundle E on X to another vector bundle E′ with rank (E′)=rank (E) and deg (E′)=deg (E)+1 (see Section 2 for its definition and its elementary properties). Using the +elementary transformations in Section 3 we will prove the following existence theorem which covers the case of a ‘small’ number of sections.

In the thesis [Pe1] it was introduced, studied and applied a general theory of Weierstrass loci for vector bundles on a smooth curve. The results, proofs, background, examples and motivations of this thesis are contained in [Pe2]. We believe that this theory, at least in characteristic 0, is the ‘right’ one. The aim of this paper is to introduce and study an extension of [Pe1] to the case of higher dimensional varieties. At least two possible theories seem to be useful and natural; see the discussion just after Remark 1.1 and Section 4. We strongly prefer the ‘symmetric’ one (see Definition 1.5). In the first section we introduce the general theory and give the main general results. In the second section we study in details the case of $P^2$ for three reasons: it is nice; it shows how to use the general theory and what could be expected in more general situations and (last but not least) to convince the reader that it is technically easier and often more interesting to work in the ‘symmetric’ set up. Then in the third section we apply the method of Section 2 to a much more general situation (essentially, any variety $X$ as base of the vector bundle). In the fourth section we give the set up and start the analysis of specific examples of what happens near a specific point $P$ of the base variety $X$ (even when $X$ is singular at $P$). Here, except at the first step we are able to work only with the ‘symmetric’ definition.

Here we study (in a more general setting) the following problem. Let C be a smooth projective curve, E and F vector bundles on C and V ⊆ H0 (C, E) (resp. W ⊆ H0 (C, F)) vector spaces generically spanning E (resp. F); find lower bounds for the dimension of the image of the multiplication map V ⊗ W → H0 (C, E ⊗ F) generalizing the case rank(E) = rank(F) = 1.

Let X be an algebraic complex projective surface equipped with the euclidean topology and E a rank 2 topological vector bundle on X. It is a classical theorem of Wu ([Wu]) that E is uniquely determined by its topological Chern classes . Viceversa, again a classical theorem of Wu ([Wu]) states that every pair (a, b) ∈ (H (X, Z), Z) arises as topological Chern classes of a rank 2 topological vector bundle. For these results the existence of an algebraic structure on X was not important; for instance it would have been sufficient to have on X a holomorphic structure. In [Sch] it was proved that for algebraic X any such topological vector bundle on X has a holomorphic structure (or, equivalently by GAGA an algebraic structure) if its determinant line bundle has a holomorphic structure. It came as a surprise when Elencwajg and Forster ([EF]) showed that sometimes this was not true if we do not assume that X has an algebraic structure but only a holomorphic one (even for some two dimensional tori (see also [BL], [BF], or [T])).

Here we give a partial classification of varieties X ⊂ Pn such that any two general zero-dimensional linear sections are projectively equivalent. They exist (with deg(X) > codim(X) + 2) only in positive characteristic.

Here we give examples and classifications of varieties with strange behaviour for the enumeration of contacts (answering a question raised by Fulton, Kleiman, MacPherson). Then we give upper and lower bounds (in terms of the degree) for the non-zero ranks of a projective variety.