Volume 10, Issue 4

We present an alternative proof of the Alexander–Hirschowitz Theorem in dimension 3 using degenerations of toric varieties.

We get a continuous one-parameter new family of embedded minimal surfaces, of which the period problems are two-dimensional. Moreover, one proves that it has Scherk's second surface and Hoffman–Wohlgemuth's example as limit-members.

We study the horospherical geometry of submanifolds in hyperbolic space. The main result is a formula for the total absolute horospherical curvature of M , which implies, for the horospherical geometry, the analogues of classical inequalities of the Euclidean Geometry. We prove the horospherical Chern–Lashof inequality for surfaces in 3-space and the horospherical Fenchel and Fary–Milnor's theorems.

Let be a log-concave function such that ƒ( x 1 , . . . , x n ) = ƒ(| x 1 |, . . . , | x n |) for every and . It was proved in [Fradelizi, Meyer, Positivity 12: 407–420, 2008] and [Fradelizi, Meyer, Adv. Math. 218: 1430–1452, 2008] that if for , , then We characterize here the case of equality: one has P(ƒ) = 4 n if and only if ƒ can be written as ƒ ( x 1 , x 2 ) = e −‖ x 1 ‖ K 1 1 K 2 ( x 2 ), where and , n 1 + n 2 = n , are unconditional convex bodies such that , i = 1, 2, where denotes the polar of K i . These last bodies were characterized in [Meyer, Israel J. Math. 55: 317–326, 1986] and [Reisner, J. London Math. Soc. 36: 126–136, 1987].

We present extensions of results of P. J. Kelly [Amer. Math. Monthly 57: 677–678, 1950] on the so-called re-entrant property of circles in strictly convex normed (or Minkowski) planes, and also further properties of circles, well known for the Euclidean plane, are generalized for all strictly convex Minkowski planes. More precisely, we present “Minkowskian analogues” of the philosophical symbol Yin-Yang, of the Arbelos, a special case of the famous Apollonius problem on circles touching each other, and one of the Sangaku-circles problems (coming from the Japanese Temple Geometry). The latter is remarkable since the consideration of this type of problems in the Euclidean plane requires the use of inversion or of the Pythagorean Theorem (i.e., of tools having no analogues in normed planes). Finally we observe that a strictly convex normed plane which, in addition, is smooth can be considered as a flat Möbius plane where, for example, Apollonius' problem is solved.

We show that the locus of stable rank four vector bundles without theta divisor over a smooth projective curve of genus two is in canonical bijection with the set of theta-characteristics. We give several descriptions of these bundles and compute the degree of the rational theta map.

In 1982/1983 A. Bichara and F. Mazzocca characterized the Grassmann space Gr( h , 𝔸) of index h of an affine space 𝔸 of dimension at least 3 over a skew-field K by means of the intersection properties of the three disjoint families of maximal singular subspaces of Gr( h , 𝔸) and, till now, their result represents the only known characterization of Gr( h , 𝔸). If K is a commutative field and 𝔸 has finite dimension m , then the image under the well known Plücker morphism is a proper subset of PG( M , K ), , called the affine Grassmannian of the h-subspaces of 𝔸. The aim of this paper is to introduce the notion of Affine Tallini Set and provide a natural and intrinsic characterization of from the point-line geometry point of view. More precisely, we prove that if a projective space over a skew-field K contains an Affine Tallini Set Δ satisfying suitable axioms on “perp” of lines, then the skew-field K is forced to be a commutative field and Δ is an affine Grassmannian, up to projections. Furthermore, several results concerning Affine Tallini Sets are stated and proved.

Let R be any commutative principal ideal domain (PID), and let n be an integer ≥ 2. Denote by 𝕀 n − 1 the set of all ( n − 1)-dimensional self-dual flats of a polarity in the projective geometry ℙ( R 2 n ). The geometric character of 𝕀 n − 1 is discussed. Two self-dual flats A, B ∈ 𝕀 n − 1 are said to be adjacent if dim( A ∩ B ) = n − 2. We prove that ϕ : 𝕀 n − 1 → 𝕀 n − 1 is an adjacency preserving surjection if and only if ϕ is an adjacency preserving bijection in both directions. Chow's theorem on the self-dual flats is extended as follows: If the polarity is the symplectic polarity and ϕ : 𝕀 n − 1 → 𝕀 n − 1 is an adjacency preserving surjection, then ϕ is induced by a collineation on ℙ( R 2 n ).

For quasihomogeneous isolated hypersurface singularities, the logarithmic comparison theorem has been characterized explicitly by Holland and Mond. In the nonquasihomogeneous case, we give a necessary condition for the logarithmic comparison theorem in terms of the Gauss–Manin system of the singularity. It shows in particular that the logarithmic comparison theorem can hold for a nonquasihomogeneous singularity only if 1 is an eigenvalue of the monodromy.

Given two surfaces in three dimensional euclidean space, one fixed and the other moved by rigid motions, we consider the total absolute curvature of the intersection curves. In this paper we investigate the integral of these total absolute curvatures over all motions. Under some geometric conditions we obtain kinematic formulas, and with weaker conditions we get upper and lower bounds. Finally, as applications, we obtain upper bounds for the average number of connected components of the intersections, and we give Hadwiger conditions for a convex domain to be able to contain another one.

Here we describe all degree n + 3 non-degenerate surfaces in , n ≥ 1, connected in codimension 1, which may be isomorphically projected into . There are three of them. One is a suitable union of n + 3 planes (for all n ≥ 1); it was discovered by Floystad. The other two are unions of a smooth quadric and two planes (only for n = 1).

Let X be a smooth projective threefold, and let A be an ample line bundle such that K X + A is nef. We show that if K X or – K X is pseudo-effective, the adjoint bundle K X + A has global sections. We also give a very short proof of the Beltrametti–Sommese conjecture in dimension three, recently proven by Fukuma: if A is an ample line bundle such that K X + 2 A is nef, the adjoint bundle K X + 2 A has global sections.