1,720,972 research outputs found
On manifolds of small degree
No full textThe main result of this paper gives a complete classification of complex smooth projective varieties X⊂Pn which are nondegenerate, linearly normal and of degree d≤n. Nondegenerate means that X is not contained in a hyperplane of Pn, and linearly normal means that the map H0(Pn,OPn(1))→H0(X,OX(1)) is surjective. The adjunction mapping plays a key role in the proof, which also relies on previous results by the author [Math. Ann. 271 (1985), no. 3, 339--348; MR0787185 (86j:14031)]. The classification has some interesting consequences. For example, X as above must be either rational or Fano, and, in particular, it is always simply connected
Birational geometry of rationally connected manifolds via quasi-lines
No full textThis is, mostly, a survey of results about the birational geometry of rationally connected manifolds, using rational curves analogous to lines in a projective space (quasi-lines)
Varieties with quadratic entry locus. II
No full textWe continue the study, begun in [F. Russo, Varieties with quadratic entry locus. I, Preprint (2006), math. AG/0701889] , of secant defective manifolds having simple entry loci. We prove that such manifolds are rational and describe them in terms of tangential projections. Using also the work of [P.Ionescu and F.Russo, Conic-connected manifolds, Preprint (2006), math. AG/0701885], their classification is reduced to the case of Fano manifolds of high index, whose Picard group is generated by the hyperplane section class. Conjecturally, the former should be linear sections of rational homogeneousmanifolds. We also provide evidence that the classification of linearly normal dual defective manifolds with Picard group generated by the hyperplane section should follow along the same lines. © 2008 Copyright Foundation Compositio Mathematica 2008
Conic-connected manifolds
No full textA smooth irreducible nondegenerate projective variety X⊂PN is said to be a conic connected manifold (CC-manifold, for short) if through two general points of X there passes an irreducible conic contained in X. CC-manifolds are in particular rationally connected.
The main theorem of the paper under review states that linearly normal CC-manifolds are Fano manifolds with second Betti number b2≤2. More precisely, it is proved that if such a CC-manifold X has b2=1 then Pic(X) is generated by the hyperplane sections and the index is at least (dimX+1)/2 except for the case X=v2(PN), the Veronese variety, and such CC-manifolds with b2=2 are completely classified: they are the inner projections from a linear subspace of the Veronese variety v2(PN), or Segre products of two projective spaces and their hyperplane sections.
A characterization of rationality via covering families of 1-cycles is also provided. For a point x on a projective variety, an x-covering family is a covering family of rational 1-cycles such that every 1-cycle from the family passes through x and a general member is smooth at x. An x-covering family is smooth if all its 1-cycles are smooth. An x-covering family is said to satisfy the infinitesimal uniqueness property if a general member of the family is uniquely determined by its tangent space at x. The characterization is the following: a projective variety X is rational if and only if for some x∈X, X admits a smooth x-covering family satisfying the infinitesimal uniqueness property. This may be interpreted as a generalization of the characterization of projective space due to K. Cho, Y. Miyaoka and N. I. Shepherd-Barron [in Higher dimensional birational geometry (Kyoto, 1997), 1--88, Adv. Stud. Pure Math., 35, Math. Soc. Japan, Tokyo, 2002; MR1929792 (2003m:14080)]
Models of rationally connected manifolds
No full textWe study rationally connected (projective) manifolds X via the concept of a model (X, Y), where Y is a smooth rational curve on X having ample normal bundle. Models are regarded from the view point of Zariski equivalence, birational on X and biregular around Y. Several numerical invariants of these objects are introduced and a notion of minimality is proposed for them. The important special case of models Zariski equivalent to (Pn, line) is investigated more thoroughly. When the (ample) normal bundle of Y in X has minimal degree, new such models are constructed via special vector bundles on X. Moreover, the formal geometry of the embedding of Y in X is analysed for some non-trivial examples. © 2003 Applied Probability Trust
Manifolds covered by lines and the Hartshorne Conjecture for quadratic manifolds
No full textSmall codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for manifolds defined by quadratic equations and to obtain the list of such Hartshorne manifolds. © 2013 by The Johns Hopkins University Press
On a theorem of van de Ven
No full textLet X⊂PN be a smooth complex connected variety. The tangent and normal bundles produce a natural exact sequence: 0→TX→TPN|X→NX/PN→0. The sequence is known to split iff X is a linear subspace.
The authors strengthen this. Let Y⊂X be an irreducible (possibly singular) curve, such that the sequence above splits, when it is restricted to Y. Then X is a linear subspace of PN.
The proof is quite easy. First, it is shown that if the restriction splits then (KX+nH,Y)X≤0. Here n=dim(X) and H is the hyperplane section. Thus KX+nH is not very ample.
Then an old result is used: If (under the conditions above) KX+nH is not very ample, then (X,H) is either (Pn,O(1)) or (Qn,O(1)) (a hyperquadric) or X is a projective bundle with the fibre being linear subspaces of PN.
Finally, the last two cases are ruled out
Remarks on defective Fano manifolds
No full textThis note continues our previous work on special secant defective (specifically, conic connected and local quadratic entry locus) and dual defective manifolds. These are now well understood, except for the prime Fano ones. Here we add a few remarks on this case, completing the results in our papers (Russo in Math Ann 344:597–617, 2009; Ionescu and Russo in Compos Math 144:949–962, 2008; Ionescu and Russo in J Reine Angew Math 644:145–157, 2010; Ionescu and Russo in Am J Math 135:349–360, 2013; Ionescu and Russo in Math Res Lett 21:1137–1154, 2014); see also the recent book (Russo, On the Geometry of Some Special Projective Varieties, Lecture Notes of the Unione Matematica Italiana, Springer, 2016)
A view on extending morphisms from ample divisors
No full textThis article studies the relation between the geometry of a smooth projective variety and that of its hyperplane sections from the viewpoint of Mori theory.
Let X be a smooth projective variety of dimension n≥4 and Y a smooth hyperplane section of X. Thus H2(Y,R)=H2(X,R) by the Lefschetz hyperplane theorem. Let p:Y→Z be a fibration of Y by Fano varieties. The authors prove several results asserting the existence of an extension of p to X under various conditions. In the main case p is the Mori contraction defined by an extremal ray R of the cone of curves NE−−−(Y) of Y in the region KY<0, and p extends iff R is also an extremal ray of NE−−−(X)
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