1,721,118 research outputs found
A few remarks on the lifting theorem.
No full textWe start with a projective variety X in Pr and a family W of projective subvarieties of Pr, parametrized by the space B, such that for any tin B the corresponding fibre Wt of W is contained in some h-plane Lt and Wt contains XnLt; furthermore, as t ranges in B, let the Lt's fill an open dense subset of the corresponding Grassmannian. We give conditions on the degrees of X, Wt which implies that the varieties Wt glue together giving a variety W (containing X) such that Wt=WnLt for all t. Our conditions, whose proves are based on the classical differential theory of "foci" introduced by C.Segre, generalize the well-known theorems of Laudal and Gruson - Peskine for the case X = curve in P3
On the Severi varieties of surfaces in P^3
No full textFor a smooth surface S in P-3 of degree d and for positive integers n, delta, the Severi variety V-n,delta(0) (S) is the subvariety of the linear system O-S(n) which parametrizes curves with delta nodes. We show that for S general, n greater than or equal to d and for all delta with 0 less than or equal to delta less than or equal to dim(O-S(n)), then V-n,delta(0)(S) has at least one component that is reduced, of the expected dimension dim(O-S(n)) - delta. We also construct examples of reducible Severi varieties on general surfaces of degree d greater than or equal to 8
The classification of (1,k)-defective surfaces
No full textWe give the full classification of surfaces X such that the family of lines contained in a (k+1)-secant P^k to X has dimension smaller than the expected
The grassmannians of secant varieties of curves are not defective
No full textWe deal with the Grassmannians of secant varieties of irreducible curves C and we show that they always have the expected dimension. This extends the well known classical result that all curves are not defective
Towards a Halphen theory of linear series on curves
No full textA linear series g^N on a curve C in P3 is primary when it does not
contain the series cut by planes. For such series, we provide a lower bound
for the degree , in terms of deg(C), g(C) and of the number s = min{i :
h^0I_C(i) > 0}. Examples show that the bound is sharp. Extensions to the
case of general linear series and to the case of curves in higher projective
spaces are considered
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