Rigid isotopy classification of real algebraic curves of bidegree (3,3) on quadrics

A rigid isotopy of nonsingular real algebraic curves on a quadric is a path in the space of such curves of a given bidegree. We obtain the rigid isotopy classification of nonsingular real algebraic curves of bidegree (3, 3) on a hyperboloid and on an ellipsoid. We also study of the space of real algebraic curves of bidegree (3, 3) with a single node or cusp. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 810–815, December, 1999.     

Asymptotic behaviour of families of real curves

We consider the family of fibres of a polynomial function f on a smooth noncompact algebraic real surface and we characterise the regular fibres of f which are atypical due to their asymptotic behaviour at infinity. We compare to the similar problem in the complex case. Received: 5 May 1998 / Revised version: 20 March 1999     

Towards relative invariants of real symplectic four-manifolds

Let (X, ω, c_X) be a real symplectic four-manifold with real part . Let be a smooth curve such that We construct invariants under deformation of the quadruple (X, ω, c_X, L) by counting the number of real rational J-holomorphic curves which realize a given homology class d, pass through an appropriate number of points and are tangent to L. As an application, we prove a relation between the count of real rational J-holomorphic curves done in [W2] and the count of reducible real rational curves done in [W3]. Finally, we show how these techniques also allow us to extract an integer valued invariant from a classical problem of real enumerative geometry, namely about counting the number of real plane conics tangent to five given generic real conics. Received: March 2005; Revision: September 2005; Accepted: September 2005     

Geometry and algebra of real forms of complex curves

Let Y be a complex algebraic curve and let be the set of all real algebraic curves with complexification , such that the real points divide . We find all such families [Y]. According to Harnak theorem a number of connected components of satisfies by the inequality , where g is the genus of Y. We prove that and these estimates are exact. Received: 15 November 2001; in final form: 28 April 2002/Published online: 2 December 2002     

Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry

We first build the moduli spaces of real rational pseudo-holomorphic curves in a given real symplectic 4-manifold. Then, following the approach of Gromov and Witten [3, 19, 11], we define invariants under deformation of real symplectic 4-manifolds. These invariants provide lower bounds for the number of real rational J-holomorphic curves which realize a given homology class and pass through a given real configuration of points. Mathematics Subject Classification (2000) 14N10, 14P25, 53D05, 53D45     

Albanese homomorphism of the Chow group of 0-cycles of a real Algebraic variety

We study a certain homomorphism of the Chow group of 0-cycles of degree zero of a real algebraic variety into the group of real points of the Albanese variety; this homomorphism is obtained from the Albanese mapping for the corresponding variety. The kernel of this homomorphism is calculated and estimates for the kernel of the mapping of the torsion groups are obtained. Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 76–83, January, 1999.     

Reduction of elliptic curves over certain real quadratic number fields

The main result of this paper is that an elliptic curve having good reduction everywhere over a real quadratic field has a -rational point under certain hypotheses (primarily on class numbers of related fields). It extends the earlier case in which no ramification at is allowed. Small fields satisfying the hypotheses are then found, and in four cases the non-existence of such elliptic curves can be shown, while in three others all such curves have been classified.

     

The complex separation of real surfaces and extensions of Rokhlin congruence

Summary The subject of this paper is the problem of topological arrangement of real algebraic curves on real algebraic surfaces. In this paper I extend Rokhlin, Kharlamov-Gudkov-Krakhnov and Kharlamov-Marin congruences and give some applications of this extension. Among these applications there are new restrictions for topological arrangement of real algebraic curves of a given degree on hyperboloid and ellipsoid, new restrictions for complex orientations of curves on a hyperboloid and the topological classification of pairs made of a non-connected cubic surfaces in P ³ and its regular intersection with a quadric.Obalatum X-1992&14-10-1993     

Invariant complex structures on solvable¶real Lie groups

In this paper, we classify the invariant complex structures on four-dimensional, solvable, simply-connected real Lie groups with commutator of dimension three. The resulting complex surfaces corresponding to these structures are also determined. The classification is based on the determination of certain complex subalgebras of the complexifications of the corresponding real Lie algebras. Received: 17 December 1999 / Revised version: 1 May 2000     

Computing the rank of elliptic curves over real quadratic number fields of class number 1

In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over . Several examples are included.

     

Welschinger invariants of toric Del Pezzo surfaces with nonstandard real structures

The Welschinger invariants of real rational algebraic surfaces are natural analogs of the Gromov-Witten invariants, and they estimate from below the number of real rational curves passing through prescribed configurations of points. We establish a tropical formula for the Welschinger invariants of four toric Del Pezzo surfaces equipped with a nonstandard real structure. Such a formula for real toric Del Pezzo surfaces with a standard real structure (i.e., naturally compatible with the toric structure) was established by Mikhalkin and the author. As a consequence we prove that for any real ample divisor D on a surface Σ under consideration, through any generic configuration of c ₁(Σ)D − 1 generic real points, there passes a real rational curve belonging to the linear system |D|. To Vladimir Igorevich Arnold on the occasion of his 70th birthday     

Totally real embeddings of the torus intoC 2

In this short note we combine a construction of Viro and a result of Eliashberg and Harlamov to prove that there exist smooth totally real embeddings of the torus intoC ² which are isotopic but not so within the class of totally real surfaces. We also show how Viro's construction can be used to define an isotopy invariant for a certain class of complex curves inC P ².     

On the fundamental homology classes of a real algebraic variety

It is proved that there is only one relation between the homology classes determined by the real points of a special real algebraic variety. This relation is equal to the sum of all the homology classes. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 216–219, August, 1999.     

Geometry of real polynomial mappings

In this paper we study the set of points at which a real polynomial mapping is not proper. Received: 7 May 1999; in final form: 18 September 2000 / Published online: 25 June 2001     

Surjectivity of power maps of real algebraic groups

In this paper we study the surjectivity of the power maps g?gn for real points of algebraic groups defined over reals. The results are also applied to study the exponentiality of such groups.     

Determination of elliptic curves with everywhere good reduction over real quadratic fields

All elliptic curves having everywhere good reduction over \Bbb Q(?{29})\Bbb Q(\sqrt {29}) are determined by studying the fields of 2- and 3-division points. As a byproduct of the argument, the elliptic curves over some real quadratic fields are determined. Though part of the result are already obtained in [2], [4], [5], [10], the proof given in the present paper is simpler.     

Separate weak*-continuity of the triple product in dual real JB*-triples

We prove that, if E is a real JB*-triple having a predual then is the unique predual of E and the triple product on E is separately $\sigma (E,E_{*_{}})-$continuous. Received February 1, 1999; in final form March 29, 1999 / Published online May 8, 2000