Threefolds in ℙ5 with One Apparent Quadruple Point

Abstract

In this article we classify all the smooth threefolds in ?⁵ with an apparent quadruple point provided that the family of its 4-secant lines is an irreducible (first order) congruence. This is sufficient to conclude the classification of all the smooth codimension two varieties in ? ⁿ with one apparent (n ? 1)-point and with irreducible family of (n ? 1)-secant lines.     

The Hilbert function of generic plane sections of curves of ℙ3

Summary A characteristic condition is given on a zero-dimensional differentiable 0-sequenceH={h _i}_i0,h ₁3, in order to be the Hilbert function of a generic plane section of a reduced irreducible curve of ³, hence of points of ² with the uniform position property. In this way an answer is given to some question stated by Harris in [Ha₂].The result is obtained by constructing a smooth irreducible arithmetically Cohen-Macaulay curve in ³ whose generic plane section has an assigned Hilbert function satisfying that condition.A.M.S. (1980) Subject classification. Primary 14C20, 14H99. Secondary 14C05     

Boundedness for Some Rationally Connected Threefolds in ℙ6

We prove boundedness of rationally-connected threefolds in ?⁶ under some extra-assumptions.     

3-Folds in ℙ5 of degree 12of degree 12

Let be a 3-dimensional submanifold of ℙ⁵ of degree 12. This article gives, up to one case, a complete classification of the deformation classes of those 3-folds. The main tools used are methods already applied in the classification of degrees 9 to 11 and adjunction theoretic results. We show here how the 2^nd reduction of can be applied to analyse the birational structure of or even exclude the existence of .     

On the Cohen-Macaulay property for monomial curves in ℙ K 3

The arithmetical Cohen-Macaulay property for monomial curves in _K ³ with generic zero was shown in [2] to be true forn ₃>(n ₂–1)(n ₂–n ₁),n ₃>n ₂, (n ₁,n ₂,n ₃)=1. Here we establish necessary and sufficient arithmetic conditions in terms ofn ₁,n ₂,n ₃ in order for the indicated curves not to be arithmetically Cohen-Macaulay.     

Lower degree curves in X3,3 ⊂ ℙ2 × ℙ2

Science China Mathematics - In this paper we study low-degree low-genus curves in a generic hypersurface X of degree (3, 3) in ?2 × ?2. We prove that the genus 0 and genus 1 curves...     

Spectrum of the Kerzman–Stein operator for a family of smooth regions in the plane

The Kerzman–Stein operator is the skew-hermitian part of the Cauchy operator defined with respect to an unweighted hermitian inner product on the boundary. For bounded regions with smooth boundary, the Kerzman–Stein operator is compact on the Hilbert space of square integrable functions. Here we give an explicit computation of its Hilbert–Schmidt norm for a family of simply connected regions. We also give an explicit computation of the Cauchy operator acting on an orthonormal basis, and we give estimates for the norms of the Kerzman–Stein and Cauchy operators on these regions. The regions are the first regions that display no apparent Möbius symmetry for which there now is explicit spectral information.     

The intersection of two quadrics in ℙ5(ℂ) as a twistor space

A parametrization is constructed for the space Y^C of conics in the intersection of two quadrics in ₅() and the study is made of the conformal structure of Y^C.     

Postulation of disjoint unions of planes and lines in ℙ5

Here we study the postulation of general disjoint unions of planes and lines in ?⁵ using the Horace Method and some unreduced structures called here 2-sundials.     

On first order congruences of lines of ℙ4 with a fundamental curve

We discuss projective families of lines of ℙ ⁿ , and in particular congruences of order one. After giving general results, we obtain a complete classification of the case of ℙ⁴ in which there is a fundamental curve. Received: 2 August 2000 / Revised version: 11 July 2001     

On the automorphisms of some line congruences in ℙ3

The automorphisms of line congruences in ³ are studied via the analysis of the automorphisms of the associated focal loci. This study is applied to a Veronese surface (i.e. to a congruence of chords of a twisted cubic) and to the rational scrolls in the Grassmannian G(1, 3).     

On algebraic points in the plane near smooth curves∗

In the paper, we solve the problem on the number of points with algebraic real coordinates near smooth curve. This question is a natural extension of the problems of number theory connected with integer points in the domains and the rational numbers near curves. The main idea of the proof is based on the metric theory on Diophantine approximations.     

Note on families of semistable curves over ℙ1 with 4 singular fibers whose Jacobian are non-compact

Let f: S → ?¹ be a semistable family of curves of genus g ? 2. We prove that if f admits exactly 5 singular fibers and 4 of them have non-compact Jacobian, then g = 2.     

Rigidity of subfamilies of hypersurfaces in ℙ n with maximal length of Griffiths-Yukawa coupling

Let f:X→U be a family of smooth hypersurfaces in P nof degree d>n+1 over a smooth curve U.Assume that the Griffiths-Yukawa coupling of f is non-vanishing.Then f is rigid.Moreover,we generalize it to the case when the Griffiths-Yukawa coupling of f is degenerated.     

Rationality of the moduli spaces of plane curves of sufficiently large degree

We prove that the moduli space of plane curves of degree d is rational for all sufficiently large d.     

The Sato–Tate distribution in families of elliptic curves with a rational parameter of bounded height

We obtain new results concerning the Sato–Tate conjecture on the distribution of Frobenius angles over parametric families of elliptic curves with a rational parameter of bounded height.     

Plane curves whose singular points are cusps and triple coverings of ℙ2

We study a plane curve C whose singular points are cusps and the surface which is a triple covering of ² branched along C. As a result, especially we obtain an inequality for the sum of the Milnor numbers at the singularities of C and new surfaces of general type.     

Two dimensional scrolls contained in quadric cones in ℙ5

Smooth, complex, ruled surfaces embedded in ℙ⁵ as linearly normal scrolls, such that they are contained in a quadric cone, are considered. Rational scrolls and some elliptic scrolls are shown to be the only ones contained in cones of rank 5. Results on scrolls contained in cones of lower ranks are also obtained.