Abstract:
Let V be a Fano threefold of genus one (i.e. with Pic(V) Z). Let G be the Grassmannian of the lines of 4 . We study when it is possible to embed V in G , in such cases we determine the cohomology class of V in H*(G).

Abstract:
Complex, smooth, projective Fano varieties were classified by Iskovskih when B2 =1 (B2 is the second Betti number) and by Mori and Mukai when B2 is at least 2. When B2 =1 it is known if such varieties are rational (unirational) or not; in this paper we solve this problem when B2 greater or equal 2.

Abstract:
We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective n-dimensional varieties, n\geq 4, onto smooth projective varieties, arising from classical projective geometry and defined over sufficiently small fields, not necessarily algebraically closed. The examples considered come from particular special homaloidal and subhomaloidal linear systems, which usually are degenerations of general phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple and Tyrrell and more recently by Ein and Shepherd-Barron. The first series of examples is associated to particular codimension 2 determinantal smooth subvarieties of P^m, 3\leq m\leq 5. We get another series of examples by considering special cubic hypersurfaces through some surfaces in P^5, or some 3-folds in P^7 having one apparent double point. The last examples come from an intriguing birational elementary extremal contraction in dimension 6, studied by Semple and Tyrrell and fully described in the last section.

Abstract:
The main goal of this paper is to give a general method to compute (via computer algebra systems) an explicit set of generators of the ideals of the projective embeddings of some ruled surfaces, namely projective line bundles over curves such that the fibres are embedded as smooth rational curves. Indeed, although the existence of the embeddings that we consider is well known, often in literature there are no explicit descriptions of the corresponding projective ideals. Such an explicit description allows to compute, besides all the syzygies, some of the important algebraic invariants of the surface, for instance the $k$-regularity, which are not always easy to compute by general formulae or by geometric arguments. An implementation of our algorithms and explicit examples for the computer algebra system Macaulay2 are included, so that anyone can use them for his own purposes.

Abstract:
Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed rational normal curves over certain linear sections of SU_C(2). This allows us to give an interpretation of these subvarieties of SUC(2) in terms of the moduli space of curves M_{0,2g}. In fact, there exists a natural linear map SU_C(2) -> P^g with modular meaning, whose fibers are birational to M_{0,2g}, the moduli space of 2g-pointed genus zero curves. If g < 4, these modular fibers are even isomorphic to the GIT compactification M^{GIT}_{0,2g}. The families of pointed rational normal curves are recovered as the fibers of the maps that classify extensions of line bundles associated to some effective divisors.

Abstract:
Very ampleness criteria for rank 2 vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.

Abstract:
We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a Hilbert scheme that parametrizes all the rational curves in $\mathbb{P}^s$ with a given decomposition type of the normal bundle and that has exactly two irreducible components. This gives a negative answer to the very old question whether such Hilbert schemes are always irreducible. We also characterize smooth non-degenerate rational curves contained in rational normal scroll surfaces in terms of the splitting type of their restricted tangent bundles, compute their normal bundles and show how to construct these curves as suitable projections of a rational normal curve.

Abstract:
Let V be a variety in P^n(C) and let W be a linear space, of dimension w, in P^n. We say that V can be isomorphically projected onto W if there exists a linear projection f, from a suitable linear space L disjoint from W, dim(L) = n-w-1 >= 0, such that f(V) is isomorphic to V. Let f' be the restriction of f to V. We say that f' is a J-embedding of V (see K. W. Johnson: Immersion and embedding of projective varieties, Acta Math. [140] (1981) 49-74) if f' is injective and the differential of f' is a finite map. In this paper we classify the J-embeddable reducible surfaces, equivalently, the reducible surfaces whose secant variety has dimension at most 4. The classification is very detailed for surfaces having two components. For three or more components we give a reasonable classification, taking into account that the complete classification is very rich of cases, subcases and subsubcases.

Abstract:
We give an explicit parametrization of the Hilbert schemes of rational curves C in P^n having a given splitting type of the restricted tangent bundle from P^n to C. The adopted technique uses the description of such curves as projections of a rational normal curve from a suitable linear vertex and a classification of those vertices that correspond to the required splitting type of the restricted tangent bundle. This classification involves the study of a suitable PGL(2) action on the relevant Grassmannian variety.