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1.
We study when double covers of P3 ramified along nodal surfaces are not Q-factorial. In particular, we describe all the Q-factorial double covers of P3 ramified along quartic surfaces with at most seven simple double points and sextic surfaces with at most 16 simple double points.  相似文献   

2.
We study the geometry of the birational map between an intersection of a net of quadrics in that contains a line and the double sextic branched along the discriminant of the net. We show that the branch locus of a smooth double sextic S 6 is discriminant of a net of quadrics in such that S 6 is isomorphic to the intersection of this net iff a certain configuration of rational curves on S6 is weakly even. Received: 14 September 2005 Suported by the DFG Schwerpunktprogramm ‘Global methods in complex geometry’. The first named author is partially supported by the KBN Grant No. 1 P03A 008 28. The second named author is partially supported by the KBN Grant No. 2 P03A 016 25.  相似文献   

3.
In any characteristic p different from 2 and 5, Kondō gave an example of a K3 surface with a purely non-symplectic automorphism of order 50. The surface was explicitly given as a double plane branched along a smooth sextic curve. In this note we show that, in any characteristic p2,5, a K3 surface with a cyclic action of order 50 is isomorphic to the example of Kondō.  相似文献   

4.
In this paper we consider a Desargues configuration in the projective plane, i.e. ten points and ten lines, on each line we have three of the points and through each point we have three of the lines. We construct a rational curve of order 6 which has a node at each of the ten points. We have never seen this kind of curve in the literature, but it is well known that for anyn there exists a rational curve of ordern which has [(n–1)(n–2)]/2 nodes and ifn=6 we find a sextic with ten nodes. The purpose of this paper is to obtain a sextic of this kind as a locus of points in connection with special projectivities of the plane associated with the Desargues configuration and to find a rational parametric representation of it. A large part of this paper is done with MACSYMA: it is an application of computer algebra in algebraic geometry. Special cases, where we find a quintic, a quartic or a cubic, are given in the last section.  相似文献   

5.
We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic surfaces.  相似文献   

6.
In a recent paper we proved a special case of a variant of Pink's Conjecture for a variety inside a semiabelian scheme: namely for any curve inside any scheme isogenous to a fibred product of two isogenous elliptic schemes. Here we go ahead with the programme of settling the conjecture for general abelian surface schemes by completing the proof for all non-simple surfaces. This involves some entirely new and crucial issues.  相似文献   

7.
In this paper we show that, for a K3 surface within a certain class of surfaces and over a number field, the orbit of a point under the group of automorphisms is either finite or its exponent of growth is exactly the Hausdorff dimension of a fractal associated to the ample cone. In particular, the exponent depends on the geometry of the surface and not its arithmetic. For surfaces in this class, the exponent is 0.6527±0.0012.  相似文献   

8.
We introduce an intrinsic property for a projective variety as follows: there exists an embedding into some projective space such that the Gauss map is of rank zero, which we call (GMRZ) for short. It turns out that (GMRZ) imposes strong restrictions on rational curves on projective varieties: In fact, using (GMRZ), we show that, contrary to the characteristic zero case, the existence of free rational curves does not imply that of minimal free rational curves in positive characteristic case. We also focus attention on Segre varieties, Grassmann varieties, and hypersurfaces of low degree. In particular, we give a characterisation of Fermat cubic hypersurfaces in terms of (GMRZ), and show that a general hypersurface of low degree does not satisfy (GMRZ).  相似文献   

9.
10.
We study the Frobenius base change of a torsor under a smooth algebraic group over a field of positive characteristic by relating it to the pushforward of the torsor under the Frobenius homomorphism. As an application, we determine the change of the multiplicity of a closed fiber of an elliptic surface by purely inseparable base changes with respect to the base curve in the case where the generic fiber is supersingular.  相似文献   

11.
12.
In this note we apply the techniques of the toric systems introduced by Hille–Perling to several problems on smooth projective surfaces: We showed that the existence of full exceptional collection of line bundles implies the rationality for small Picard rank surfaces; we proved equivalences of several notions of cyclic strong exceptional collection of line bundles; we also proposed a partial solution to a conjecture on exceptional sheaves on weak del Pezzo surfaces.  相似文献   

13.
Here we define the concept of Qregularity for coherent sheaves on a smooth quadric hypersurface QnPn+1. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on Qn with the Castelnuovo-Mumford regularity of their extension by zero in Pn+1. We also classify the coherent sheaves with Qregularity −. We use our notion of Qregularity in order to prove an extension of the Evans-Griffiths criterion to vector bundles on quadrics. In particular, we get a new and simple proof of Knörrer’s characterization of ACM bundles.  相似文献   

14.
In this paper we study the Hodge numbers of a branched double covering of a smooth, complete algebraic threefold. The involution on the double covering gives a splitting of the Hodge groups into symmetric and skew-symmetric parts. Since the symmetric part is naturally isomorphic to the corresponding Hodge group of the base we study only the skew-symmetric parts and prove that in many cases it can be computed explicitly. Received: 6 March 2001 / in final form: 4 September 2001/ Published online: 4 April 2002  相似文献   

15.
Let E be an elliptic curve over an infinite field K with characteristic ≠2, and σH1(GK,E)[2] a two-torsion element of its Weil-Châtelet group. We prove that σ is always visible in infinitely many abelian surfaces up to isomorphism, in the sense put forward by Cremona and Mazur in their article (J. Exp. Math. 9(1) (2000) 13). Our argument is a variant of Mazur's proof, given in (Asian J. Math. 3(1) (1999) 221), for the analogous statement about three-torsion elements of the Shafarevich-Tate group in the setting where K is a number field. In particular, instead of the universal elliptic curve with full level-three-structure, our proof makes use of the universal elliptic curve with full level-two-structure and an invariant differential.  相似文献   

16.
We conjecture that derived categories of coherent sheaves on fake projective n  -spaces have a semi-orthogonal decomposition into a collection of n+1n+1 exceptional objects and a category with vanishing Hochschild homology. We prove this for fake projective planes with non-abelian automorphism group (such as Keum's surface). Then by passing to equivariant categories we construct new examples of phantom categories with both Hochschild homology and Grothendieck group vanishing.  相似文献   

17.
We prove that for any of a wide class of elliptic surfaces X defined over a number field k, if there is an algebraic point on X that lies on only finitely many rational curves, then there is an algebraic point on X that lies on no rational curves. In particular, our theorem applies to a large class of elliptic K3 surfaces, which relates to a question posed by Bogomolov in 1981.  相似文献   

18.
Let f:X→Yf:XY be a morphism between normal complex varieties, where YY is Kawamata log terminal. Given any differential form σσ, defined on the smooth locus of YY, we construct a “pull-back form” on XX. The pull-back map obtained by this construction is ?Y?Y-linear, uniquely determined by natural universal properties and exists even in cases where the image of ff is entirely contained in the singular locus of YY.  相似文献   

19.
Let V be a reduced and irreducible hypersurface of degree k 3. In this paper we prove that if the singular locus of V consists of 2 ordinary double points, 3 ordinary triple points and if 2 + 43 < (k – 1)2, then any smooth surface contained in V is a complete intersection on V.Received: 7 January 2004  相似文献   

20.
We prove the existence of hypersurfaces defined over finite fields having a prescribed number of -rational points and a prescribed number of non-singular points. Moreover, some results on -rational intersections between plane curves, lines and conics, are given. Received: June 1, 2006. Revised: August 1, 2007.  相似文献   

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