A new characterization of projections of quadrics in finite projective spaces of even characteristic

We will classify, up to linear representations, all geometries fully embedded in an affine space with the property that for every antiflag {p,L} of the geometry there are either 0, α, or q lines through p intersecting L. An example of such a geometry with α=2 is the following well known geometry . Let Q_n+1 be a nonsingular quadric in a finite projective space , n≥3, q even. We project Q_n+1 from a point r∉Q_n+1, distinct from its nucleus if n+1 is even, on a hyperplane not through r. This yields a partial linear space whose points are the points p of , such that the line 〈p,r〉 is a secant to Q_n+1, and whose lines are the lines of which contain q such points. This geometry is fully embedded in an affine subspace of and satisfies the antiflag property mentioned. As a result of our classification theorem we will give a new characterization theorem of this geometry.     

Minimal Covers of Q +(2n + 1, q) by (n − 1)-Dimensional Subspaces

A t-cover of a quadric is a set C of t-dimensional subspaces contained in such that every point of is contained in at least one element of C.We consider (n – 1)-covers of the hyperbolic quadric Q ⁺(2n + 1, q). We show that such a cover must have at least q ^{n + 1} + 2q + 1 elements, give an example of this size for even q and describe what covers of this size should look like.     

Real Hypersurfaces Having Many Pseudo-hyperplanes

Abstract

Let nand dbe natural integers satisfying n ≥ 3 and d ≥ 10. Let Xbe an irreducible real hypersurface Xin ? ⁿ of degree dhaving many pseudo-hyperplanes. Suppose that Xis not a projective cone. We show that the arrangement ? of all d ? 2 pseudo-hyperplanes of Xis trivial, i.e., there is a real projective linear subspace Lof ? ⁿ (?) of dimension n ? 2 such that L ? Hfor all H ∈ ?. As a consequence, the normalization of Xis fibered over ?¹in quadrics. Both statements are in sharp contrast with the case n = 2; the first statement also shows that there is no Brusotti-type result for hypersurfaces in ? ⁿ , for n ≥ 3.     

二次曲面和平面位置关系的判式

在解析几何中有二次曲线与直线位置关系的讨论、二次曲面与直线位置关系的讨论,而二次曲面与平面相关位置关系的探讨较少.本文给出二次曲面a11x2+a22y2+a33z2+2a12xy+2a13xz+2a23yz+2a14x+2a24y+2a34z+a44=0(1)和平面Ax+By+Cz+D=0(2)的相对位置的判别式Δ=a11a12a13a14Aa21a22a23a24Ba31a32a33a34Ca41a42a43a44DA B C D0(aij=aji).(3)并证明了:若Δ>0,则二次曲面(1)与平面(2)相交;若Δ=0,则(1)和(2)相切;若Δ<0,则(1)和(2)相离.     

A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics

We prove that there are only finitely many families of codimension two nonsingular subvarieties of quadrics which are not of general type, for and . We prove a similar statement also for the case of higher codimension.

     

惯性定律的新证法

本文从实对称矩阵的角度 ,给出了惯性定律的一种新证法 .这种证法可使我们从已给的一个实二次型 f( x1 ,x2 ,… ,xn) =∑ni,j=1aijxixj 的矩阵 A=( aij) n× n的诸元素 ,来直接研究这个二次型的性质的这一体系更加完整 .同时 ,本文还大大地改进了雅可比方法 ,使雅可比的方法更加完美 ,应用更加广泛 .     

Gröbner Flags and Gorenstein Algebras

The goal of this paper is to study the Koszul property and the property of having a Gröbner basis of quadrics for classical varieties and algebras as canonical curves, finite sets of points and Artinian Gorenstein algebras with socle in low degree. Our approach is based on the notion of Gröbner flags and Koszul filtrations. The main results are the existence of a Gröbner basis of quadrics for the ideal of the canonical curve whenever it is defined by quadrics, the existence of a Gröbner basis of quadrics for the defining ideal of s 2n points in general linear position in P ⁿ , and the Koszul property of the generic Artinian Gorenstein algebra of socle degree 3.