On a theorem of Castelnuovo and applications to moduli

In this paper we prove a theorem stated by Castelnuovo which bounds the dimension of linear systems of plane curves in terms of two invariants, one of which is the genus of the curves in the system. Then we classify linear systems whose dimension belongs to certain intervals which naturally arise from Castelnuovo's theorem. Finally we make an application to the following moduli problem: what is the maximum number of moduli of curves of geometric genus $g$ varying in a linear system on a surface? It turns out that, for $g\ge 22$, the answer is $2g+1$, and it is attained by trigonal canonical curves varying on a balanced rational normal scroll.Comment: 8 page

Birational classification of curves on rational surfaces

In this paper we consider the birational classification of pairs (S,L), with S a rational surfaces and L a linear system on S. We give a classification theorem for such pairs and we determine, for each irreducible plane curve B, its "Cremona minimal" models, i.e. those plane curves which are equivalent to B via a Cremona transformation, and have minimal degree under this condition.Comment: 33 page

Brill--Noether loci of stable rank--two vector bundles on a general curve

In this note we give an easy proof of the existence of generically smooth components of the expected dimension of certain Brill--Noether loci of stable rank 2 vector bundles on a curve with general moduli, with related applications to Hilbert scheme of scrolls.Comment: 9 pages, submitted preprin

On Cremona contractibility of unions of lines in the plane

We discuss the concept of Cremona contractible plane curves, with an historical account on the development of this subject. We then classify Cremona contractible unions of d > 11 lines in the plane.Comment: 14 pages; removed section 5 which contained an incomplete proof; accepted for publication on Kyoto Journal of Mathematic

Scrolls and hyperbolicity

Using degeneration to scrolls, we give an easy proof of non-existence of curves of low genera on general surfaces in P3 of degree d >=5. We show, along the same lines, boundedness of families of curves of small enough genera on general surfaces in P3. We also show that there exist Kobayashi hyperbolic surfaces in P3 of degree d = 7 (a result so far unknown), and give a new construction of such surfaces of degree d = 6. Finally we provide some new lower bounds for geometric genera of surfaces lying on general hypersurfaces of degree 3d > 15 in P4.Comment: 17