Polynomial-Time Computation of the Degree of a Dominant Morphism in Characteristic Zero. I

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables over the field of characteristic zero. We show how to compute the degree of a dominant rational morphism from W to W′. The morphism is given by homogeneous polynomials of degree d′.This algorithms is deterministic and polynomial in (dd′)ⁿ and the size of the input. Bibliography: 11 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2003, pp. 189–235.     

Polynomial-time computation of the degree of a dominant morphism in zero characteristic. III

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables in zero characteristic. Consider a dominant rational morphism from W to W′ given by homogeneous polynomials of degree d′. We suggest algorithms for constructing objects in general position related to this morphism. These algorithms are deterministic and polynomial in (dd′)ⁿ and the size of the input. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 344, 2007, pp. 203–239.     

Remarks on the Gieseker-Petri divisor in genus eight

In the moduli space of curves of genus 8, M ₈, denote by GP ₈ the locus of curves that do not satisfy the Gieseker-Petri theorem. In this short note we study the projective plane models of curves of genus 8 that do not satisfy the Gieseker-Petri theorem. We use these projective models to exhibit an irreducible divisorial component in GP ₈ and we show that GP ₈ is an irreducible divisor.     

Immersions of the projective plane with one triple point

We consider C^∞ generic immersions of the projective plane into the 3-sphere. Pinkall has shown that every immersion of the projective plane is homotopic through immersions to Boy's immersion, or its mirror. There is another lesser-known immersion of the projective plane with self-intersection set equivalent to Boy's but whose image is not homeomorphic to Boy's. We show that any C^∞ generic immersion of the projective plane whose self-intersection set in the 3-sphere is connected and has a single triple point is ambiently isotopic to precisely one of these two models, or their mirrors. We further show that any generic immersion of the projective plane with one triple point can be obtained by a sequence of toral and spherical surgical modifications of these models. Finally we present some simple applications of the theorem regarding discrete ambient automorphism groups; image-homology of immersions with one triple point; and almost tight ambient isotopy classes.     

Into how many regions do <Emphasis Type="Italic">n</Emphasis> lines divide the plane if at most <Emphasis Type="Italic">n − k</Emphasis> of them are concurrent?

The number of connected components of the complement in the real projective plane to a family of n ≥2 different lines such that any point belongs to at most n − k of them is estimated. If $ n \geqslant \frac{{k^2 + k}} {2} + 3 $ n \geqslant \frac{{k^2 + k}} {2} + 3 , then the number of regions is at least (k+1)(n−k). Thus, a new proof of N. Martinov’s theorem is obtained. This theorem determines all pairs of integers (n, f) such that there is an arrangement of n lines dividing the projective plane into f regions.     

On various types of homogeneous Riemannian spaces with an isotropy group which decomposes

Homogeneous Riemannian spaces are considered whose isotropy group H decomposes into the direct product of irreducible subgroups and the identity operator acting in mutually orthogonal planes in the tangent space of a point M. We exclude the special cases when an irreducible subgroup in the decomposition of H is semisimple and acts on a plane whose dimension is a multiple of four. These spaces admit a rigid tensor structuref satisfying the conditionf ³ +f = 0.Translated from Matematicheskie Zametki, Vol. 5, No. 3, pp. 361–372, March, 1969.     

Stabilizers of collineation groups of smooth stable planes

Let be a smooth stable plane of dimension n (see Definition 1.2) and let Δ be a closed subgroup of the collineation group of which fixes some point p. We derive some results on the group-theoretical structure of Δ, e.g. that Δ is a linear Lie group (Theorem 3.7). As a by-product this shows that no (affine or projective) Moulton plane can be turned into a smooth plane. If Δ fixes some flag, then any Levi subgroup Ψ of Δ is a compact group and Δ is contained in the flag stabilizer of the classical Moufang plane of dimension n (Corollary 3.1 and Theorem 3.7). Let Δ fix three concurrent lines through the point p. If is one of the classical projective planes over the reals, the complex numbers, the quaternions, or the Cayley numbers, then the dimension of Δ is d_class = 3, 6, 15, or 38, respectively. We show that for a smooth stable (projective) plane S of dimension 2l either S is an almost projective translation plane (classical projective plane) or that dim Δ ≤ d_class − l holds (Theorems 4.1 and 4.2).     

Polynomial-time computation of the degree of a dominant morphism in zero characteristic. II

Consider a projective algebraic variety W which is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables over a field of zero characteristic. Consider a dominant rational morphism from W to W′ given by homogeneous polynomials of degree d′. We suggest algorithms for constructing objects in general position related to this morphism. They generalize some algorithms from the first part of the paper to the case dim W > dim W′. These algorithms are deterministic and polynomial in (dd′)n and the size of the input. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 181–224.     

Polynomial-time computation of the degree of a dominant morphism in zero characteristic. IV

Consider a projective algebraic variety W that is an irreducible component of the set of all common zeros of a family of homogeneous polynomials of degrees less than d in n + 1 variables in zero characteristic. Consider a dominant rational morphism from W to W′ given by homogeneous polynomials of degree d′. We suggest algorithms for constructing objects in general position related to this morphism. These algorithms are deterministic and polynomial in (dd′) ⁿ and the size of the input. This work concludes a series of four papers. Bibliography: 13 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 360, 2008, pp. 260–294.     

Subgraphs with Restricted Degrees of their Vertices in Large Polyhedral Maps on Compact Two-manifolds

Let k ≥ 2, be an integer and M be a closed two-manifold with Euler characteristic χ(M) ≤ 0. We prove that each polyhedral map G onM , which has at least (8 k² + 6 k − 6)|χ (M)| vertices, contains a connected subgraph H of order k such that every vertex of this subgraph has, in G, the degree at most 4 k + 4. Moreover, we show that the bound 4k + 4 is best possible. Fabrici and Jendrol’ proved that for the sphere this bound is 10 ifk = 2 and 4 k + 3 if k ≥ 3. We also show that the same holds for the projective plane.     

Moduli of symplectic instanton vector bundles of higher rank on projective space ℙ<Superscript>3</Superscript>

Symplectic instanton vector bundles on the projective space ℙ³ constitute a natural generalization of mathematical instantons of rank-2. We study the moduli space I _n;r of rank-2r symplectic instanton vector bundles on ℙ³ with r ≥ 2 and second Chern class n ≥ r, n ≡ r (mod 2). We introduce the notion of tame symplectic instantons by excluding a kind of pathological monads and show that the locus I _n;r ^* of tame symplectic instantons is irreducible and has the expected dimension, equal to 4n(r + 1) −r(2r + 1).     

Flex Curves and their Applications

In this paper, we define the notion of the flex curve F ⁽⁾(f; P) at a nonsingular point P of a plane curve Ca. We construct interesting plane curves using a cyclic covering transform, branched along F ⁽⁾(f; P). As an application, we show the moduli space of projective curves of degree 12 with 27 cusps has at least three irreducible components. Simultaneously, we give an example of Alexander-equivalent Zariski pair of irreducible curves.     

On a problem of Erdős and Larson

It is shown that a pairwise balanced design onn points in which each block is of size at leastn ^1/2 −c can be embedded in a projective plane of ordern+i for somei≦c + 2 ifn is sufficiently large. Among other things this implies that if the projective plane conjecture is true, the conjecture of Erdős and Larson will not be true. The paper was written during a visit of the second author to Mehta Research Institute, Allahabad.     

Generating even triangulations of the projective plane

We shall determine the 20 families of irreducible even triangulations of the projective plane. Every even triangulation of the projective plane can be obtained from one of them by a sequence of even‐splittings and attaching octahedra, both of which were first given by Batagelj 2 . © 2007 Wiley Periodicals, Inc. J Graph Theory 56: 333–349, 2007     

Minimal Cubic Cones via Clifford Algebras

In this paper, we construct two infinite families of algebraic minimal cones in ^n{\mathbb{R}^{n}}. The first family consists of minimal cubics given explicitly in terms of the Clifford systems. We show that the classes of congruent minimal cubics are in one to one correspondence with those of geometrically equivalent Clifford systems. As a byproduct, we prove that for any n ≥ 4, n ≠ 16k + 1, there is at least one minimal cone in \mathbbRⁿ{\mathbb{R}^{n}} given by an irreducible homogeneous cubic polynomial. The second family consists of minimal cones in \mathbbR^m2{\mathbb{R}^{m^2}}, m ≥ 2, defined by an irreducible homogeneous polynomial of degree m. These examples provide particular answers to the questions on algebraic minimal cones in \mathbbRⁿ{\mathbb{R}^{n}} posed by Wu-Yi Hsiang in the 1960s.     

Modular irreducible representations of the symmetric group as linear codes

We describe a particularly easy way of evaluating the modular irreducible matrix representations of the symmetric group. It shows that Specht’s approach to the ordinary irreducible representations, along Specht polynomials, can be unified with Clausen’s approach to the modular irreducible representations using symmetrized standard bideterminants. The unified method, using symmetrized Specht polynomials, is very easy to explain, and it follows directly from Clausen’s theorem by replacing the indeterminate x_ij of the letter place algebra by x_jⁱ.Our approach is implemented in SYMMETRICA. It was used in order to obtain computational results on code theoretic properties of the p-modular irreducible representation [λ]_p corresponding to a p-regular partition λ via embedding it into representation spaces obtained from ordinary irreducible representations. The first embedding is into the permutation representation induced from the column group of a standard Young tableau of shape λ. The second embedding is the embedding of [λ]_p into the space of , the p-modular representation obtained from the ordinary irreducible representation [λ] by reducing the coefficients modulo p.We include a few tables with dimensions and minimum distances of these codes; others can be found via our home page.     

A remark on the uniqueness of embeddings of linear spaces into desarguesian projective planes

Suppose that q ? 2 is a prime power. We show that a linear space with a(q + 1)² + (q + 1) points, where a ? 0.763, can be embedded in at most one way in a desarguesian projective plane of order q. © 1995 John Wiley & Sons, Inc.     

Hjelmslevgruppen mit Nachbarhomomorphismus

Hjelmslev groups have been introduced by F. Bachmann ([1], [2]) in order to study plane metric geometries in a general sense: For example two points may have none or two lines joining them. Let (G,S) and (-G,¯ S) be Hjelmslev groups and let be a Hjelmslev homomorphism from (G,S) onto (¯G, ¯S). It is shown that — under certain assumptions — the group plane of (G, S) can be embedded into the projective Hjelmslev plane over a local ringR and thatG is isomorphic to a subgroup of an orthogonal group O ₃ ⁺ (V,f). The result may be considered as a generalization of the main theorem in Bachmann [1].     

Intersection of Curves and Crossing Number of C m × C n on Surfaces

Let be two families of closed curves on a surface , such that , each curve in intersects each curve in , and no point of is covered three times. When is the plane, the projective plane, or the Klein bottle, we prove that the total number of intersections in is at least 10mn/9 , 12mn/11 , and mn+10 ^-13 m ² , respectively. Moreover, when m is close to n , the constants are improved. For instance, the constant for the plane, 10/9 , is improved to 8/5 , for n ≤ 5(m-1)/4 . Consequently, we prove lower bounds on the crossing number of the Cartesian product of two cycles, in the plane, projective plane, and the Klein bottle. All lower bounds are within small multiplicative factors from easily derived upper bounds. No general lower bound has been previously known, even on the plane. Received January 20, 1996, and in revised form October 21, 1996.     

On surfaces and their contours

We prove two theorems concerning the global behaviour of a smooth compact surfaceS, without boundary, embedded in a real projective space or mapped to a plane. Our starting point is an analysis of the orientability properties of the normal bundle of a singular projective curve. Then we see how an excellent projection fromS to the Euclidean plane gives rise to integral relations linking the singularities of the apparent contour. Finally, given an embedding ofS in RPⁿ, we look at the discriminant Δ^* of a net of hyperplanes that intersectsS in a generic way, obtaining a characterization of Δ^* in terms of mod.2 cohomology invariants.