142 research outputs found
On the classification of defective threefolds
We classify all irreducible projective threefolds which are
-defective, i.e. some -secant variety of has dimension less than the
expected value. This results extends the classical Scorza's classification of
the case .Comment: AMSLaTeX, 30 page
On the Severi varieties of surfaces in P^3
The Severi variety V_{n,d} of a smooth projective surface S is defined as the
subvariety of the linear system |O_S(n)|, which parametrizes curves with d
nodes. We show that, for a general surface S of degree k in P^3 and for all
n>k-1, d=0,...,dim(|O_S(n)|), there exists one component of V_{n,d} which is
reduced, of the expected dimension dim(|O_S(n)|)-d. Components of the expected
dimension are the easiest to handle, trying to settle an enumerative geometry
for singular curves on surfaces. On the other hand, we also construct examples
of reducible Severi varieties, on general surfaces of degree k>7 in P^3.Comment: AMSTeX, AMSppt style, 14 page
Halphen conditions and postulation of nodes
We give sharp lower bounds for the postulation of the nodes of a general
plane projection of a smooth connected curve C in P^r and we study the
relationships with the geometry of the embedding. Strict connections with
Castelnuovo's theory and Halphen's theory are shown.Comment: LaTeX, 26 page
Subvarieties of generic hypersurfaces in any variety
Let W be a projective variety of dimension n+1, L a free line bundle on W, X
in a hypersurface of degree d which is generic among those given by
sums of monomials from , and let be a generically finite map
from a smooth m-fold Y. We suppose that f is r-filling, i.e. upon deforming X
in , f deforms in a family such that the corresponding deformations
of dominate . Under these hypotheses we give a lower bound for the
dimension of a certain linear system on the Cartesian product having
certain vanishing order on a diagonal locus as well as on a double point locus.
This yields as one application a lower bound on the dimension of the linear
system |K_{Y} - (d - n + m)f^*L - f^*K_{W}| which generalizes results of Ein
and Xu (and in weaker form, Voisin). As another perhaps more surprising
application, we conclude a lower bound on the number of quadrics containing
certain projective images of Y.Comment: We made some improvements in the introduction and definitions. In an
effort to clarify the arguments we separated the 1-filling case from the
r-filling case and we gave a more detailed proof of the key lemma. The
article will appear in the Math. Proc. Cambridge Philos. So
Terracini loci of curves
We study subsets S of curves X whose double structure does not impose independent conditions to a linear series L, but there are divisors D∈ | L| singular at all points of S. These subsets form the Terracini loci of X. We investigate Terracini loci, with a special look towards their non-emptiness, mainly in the case of canonical curves, and in the case of space curves
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