Unirationality of certain supersingular K3 surfaces in characteristic 5

We show that every supersingular K3 surface in characteristic 5 with Artin invariant ≤ 3 is unirational.     

Instability of certain capillary surfaces

A family of capillary surface configurations, shown in an earlier paper to be unstable for sufficiently small Bond numberB, is here shown to be unstable also forB sufficiently large. Numerical evaluations indicate that it is unstable for allB. Some numerical calculations of the corresponding container shapes are included.     

Nonrigidity of certain composite surfaces of revolution

The nonrigidity of the composite surfaces of revolution 2 = S₁ + S₂+ S₃ is analyzed, where S₁ and S₂ are internally glued and S₂ and S₃ are externally glued together. It is shown that cases of rigidity as well as nonrigidity can obtain for surfaces of this type.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 333–344, September, 1971.The author is grateful to I. Kh. Sabitov for suggesting the topic and directing the study.     

On certain surfaces of revolution of the second degree

An improvement in the generation of some second-order surfaces of rotation and their construction is presented. In this case, a cube is rotated about one its diagonals to obtain a hyperboloid of one sheet with two duplicate cones on each side.     

Zero cycles on certain surfaces in arbitrary characteristic

Letk be a field of arbitrary characteristic. LetS be a singular surface defined overk with multiple rational curve singularities and suppose that the Chow group of zero cycles of its normalisation is finite dimensional. We give numerical conditions under which the Chow group of zero cycles ofS is finite dimensional.     

Nonexistence of certain complete minimal surfaces with planar ends

In this paper we prove that two lines bounding an immersed minimal surface in a slab in R ³ homeomorphic to a compact Riemann surface minus two disks and a finite number of points must be parallel. This theorem is extended to a higher dimensional minimal hypersurface. Also it is proved that if the Gauss map of a complete embedded minimal surface of finite total curvature at a planar end has order two, then the intersection of the surface with the plane asymptotic to the planar end cannot admit a one-to-one orthogonal projection onto any line in the plane. Received: November 26, 1998     

Free resolutions of the defining ideal of certain rational surfaces

We consider the surface obtained from the projective plane by blowing up the points of intersection of two plane curves of same degree meeting transversely. We find minimal free resolutions of the defining ideals of these surfaces embedded in projective space by the sections of a very ample divisor class. All of the results are proven over an algebraically closed field of arbitrary characteristic.     

Maximal surfaces in a certain 3-dimensional homogeneous spacetime

A generalized integral representation formula for spacelike maximal surfaces in a certain 3-dimensional homogeneous spacetime is obtained. This spacetime has a solvable Lie group structure with left invariant metric. The normal Gauß map of maximal surfaces in the homogeneous spacetime is discussed and the harmonicity of the normal Gauß map is studied.     

Positivity of certain functions associated with analysis on elliptic surfaces

In this paper, we study functions of one variable that are called boundary terms of two-dimensional zeta integrals established in recent works of Ivan Fesenko?s two-dimensional adelic analysis attached to arithmetic elliptic surfaces. It is known that the positivity of the fourth log derivatives of boundary terms around the origin is a sufficient condition for the Riemann hypothesis of Hasse-Weil L-functions of elliptic curves. We show that such positivity is also a necessary condition under some reasonable technical assumptions.     

The Hilbert function of curves on certain smooth quartic surfaces

The Riemann-Roch problem for divisors on a smooth surface in ³ is studied. This problem is solved for some smooth quartic surfaces which are called Mori quartics; as a consequence the Hilbert function of any integral curve on a Mori quartic is determined.     

On certain Mordell-Weil lattices of hyperelliptic type on rational surfaces

We study a class of rational hyperelliptic fibrations that can serve as a natural higher-genus analogue of rational elliptic fibrations from the standpoint of Mordell-Weil lattices. As a corollary, we obtain certain generalizations of the root lattices. We also describe the torsion part. Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory. Vol. 62, Algebraic Geometry-10, 1999.