Hyperplane Sections Research Articles

Projective curves with maximal regularity and applications to syzygies and surfaces

We first show that the union of a projective curve with one of its extremal secant lines satisfies the linear general position principle for hyperplane sections. We use this to give an improved approximation of the Betti numbers of curves $${{\mathcal C}\subset \mathbb P^r_K}$$ of maximal regularity with $${{\rm deg}\, {\mathcal C}\leq 2r -3}$$ . In particular we specify the number and degrees of generators of the vanishing ideal of such curves. We apply these results to study surfaces $${X \subset \mathbb P^r_K}$$ whose generic hyperplane section is a curve of maximal regularity. We first give a criterion for “an early descent of the Hartshorne-Rao function” of such surfaces. We use this criterion to give a lower bound on the degree for a class of these surfaces. Then, we study surfaces $${X \subset\mathbb P^r_K}$$ for which $${h^1(\mathbb P^r_K, {\mathcal I}_X(1))}$$ takes a value close to the possible maximum deg X − r + 1. We give a lower bound on the degree of such surfaces. We illustrate our results by a number of examples, computed by means of Singular, which show a rich variety of occuring phenomena.

Modular properties of nodal curves on K3 surfaces

In this article we are going to address the following issues: (1) the first is a rigidity property for pairs (S, C) consisting of a general projective K3 surface S, and a curve C obtained as the normalization of a nodal, hyperplane section \({\hat C\hookrightarrow S}\). We prove that a non-trivial deformation of a pair (S, C) induces a non-trivial deformation of C; (2) the second question concerns the Wahl map of curves C obtained as above. We prove that the Wahl map of the normalization of a nodal curve contained in a general projective K3 surface is non-surjective. In both cases, we impose upper bounds on the number of nodes of \({\hat C}\).

Face rings of simplicial complexes with singularities

The face ring of a simplicial complex modulo m generic linear forms is shown to have finite local cohomology if and only if the link of every face of dimension m or more is nonsingular, i.e., has the homology of a wedge of spheres of the expected dimension. This is derived from an enumerative result for local cohomology of face rings modulo generic linear forms, as compared with local cohomology of the face ring itself. The enumerative result is generalized to squarefree modules. A concept of Cohen–Macaulay in codimension c is defined and characterized for arbitrary finitely generated modules and coherent sheaves. For the face ring of an r-dimensional complex Δ, it is equivalent to nonsingularity of Δ in dimension r − c; for a coherent sheaf on projective space, this condition is shown to be equivalent to the same condition on any single generic hyperplane section. The characterization of nonsingularity in dimension m via finite local cohomology thus generalizes from face rings to arbitrary graded modules.

Estimates for the extremal sections of [formula omitted]-balls

The problem of finding the maximal hyperplane section of Bpn, where p>2, has been open for a long time. It is known that the answer depends on both p and n. In this paper, using the well-known equivalence between hyperplane sections and the isotropic constant of a body, we give an upper bound estimate for the volume of hyperplane sections of normalized ℓpn-balls that does not depend on n and p. In addition, on the basis of results of Meyer, Pajor and Schmuckenschläger, we show further the corresponding extremal body and hyperplane section when this volume attains its minimum.

On the Debarre-de Jong and Beheshti-Starr conjectures on hypersurfaces with too many lines

Conjecture 1.1. Let Xn−1 ⊂ P n be a hypersurface of degree d ≥ n and let F(X) ⊂ G(2, n+ 1) denote the Fano scheme of lines on X. Let B ⊂ F(X) be an irreducible component of dimension at least n − 2. Let IB := {(x,E) | x ∈ X, E ∈ B, x ∈ PE}, and let π and ρ denote (respectively) the projections to X and B. Let XB = π(IB) ⊆ X and let Cx = πρ−1ρπ−1(x). Then, for all x ∈XB , Cx ∩Xsing = ∅. If we take hyperplane sections in the case d = n, then Conjecture 1.1 would imply the following, which was conjectured independently by Debarre and de Jong.

Configurations of points and strings

Let X be a smooth projective variety of dimension n ≥ 2 . It is shown that a finite configuration of points on X subject to certain geometric conditions possesses rich inner structure. On the mathematical level this inner structure is a variation of Hodge-like structure. As a consequence one can attach to such point configurations: (i) Lie algebras and their representations; (ii) a Fano toric variety whose hyperplane sections are Calabi–Yau varieties. These features imply that the points cease to be zero-dimensional objects and acquire dynamics of linear operators “propagating” along the paths of a particular trivalent graph. Furthermore, following particular linear operators along the “shortest” paths of the graph, one creates, for every point of the configuration, a distinguished hyperplane section of the Fano variety in (ii), i.e. the points “open up” to become Calabi–Yau varieties. Thus one is led to a picture which is very suggestive of quantum gravity according to string theory.

On the second Gaussian map for curves on a K3 surface

AbstractBy a theorem of Wahl, for canonically embedded curves which are hyperplane sections of K3 surfaces, the first Gaussian map is not surjective. In this paper we prove that if C is a general hyperplane section of high genus (> 280) of a general polarized K3 surface, then the second Gaussian map of C is surjective. The resulting bound for the genus g of a general curve with surjective second Gaussian map is decreased to g > 152.

On the second Gaussian map for curves on a K3 surface

AbstractBy a theorem of Wahl, for canonically embedded curves which are hyperplane sections of K3 surfaces, the first Gaussian map is not surjective. In this paper we prove that ifCis a general hyperplane section of high genus (> 280) of a general polarized K3 surface, then the second Gaussian map ofCis surjective. The resulting bound for the genusgof a general curve with surjective second Gaussian map is decreased tog> 152.

Triple-point defective surfaces

Abstract In this paper we study the linear series |L – 3p| of hyperplane sections with a triple point p on a surface S embedded via a very ample line bundle L for a general point p. If this linear series does not have the expected dimension we call (S, L) triple-point defective. We show that on a triple-point defective surface through a general point every hyperplane section has either a triple component or the surface is rationally ruled and the hyperplane section contains twice a fibre of the ruling.

On the integrability of symplectic Monge–Ampère equations

Let u be a function of n independent variables x 1 , … , x n , and let U = ( u i j ) be the Hessian matrix of u . The symplectic Monge–Ampère equation is defined as a linear relation among all possible minors of U . Particular examples include the equation det U = 1 governing improper affine spheres and the so-called heavenly equation, u 13 u 24 − u 23 u 14 = 1 , describing self-dual Ricci-flat 4 -manifolds. In this paper we classify integrable symplectic Monge–Ampère equations in four dimensions (for n = 3 the integrability of such equations is known to be equivalent to their linearisability). This problem can be reformulated geometrically as the classification of ‘maximally singular’ hyperplane sections of the Plücker embedding of the Lagrangian Grassmannian. We formulate a conjecture that any integrable equation of the form F ( u i j ) = 0 in more than three dimensions is necessarily of the symplectic Monge–Ampère type.

GENERALIZED ZARISKI–VAN KAMPEN THEOREM AND ITS APPLICATION TO GRASSMANNIAN DUAL VARIETIES

We formulate and prove a generalization of Zariski–van Kampen theorem on the topological fundamental groups of smooth complex algebraic varieties. As an application, we prove a hyperplane section theorem of Lefschetz–Zariski–van Kampen type for the fundamental groups of the complements to the Grassmannian dual varieties.

Stability of the Blaschke–Santaló and the affine isoperimetric inequality

A stability version of the Blaschke–Santaló inequality and the affine isoperimetric inequality for convex bodies of dimension n ⩾ 3 is proved. The first step is the reduction to the case when the convex body is o-symmetric and has axial rotational symmetry. This step works for related inequalities compatible with Steiner symmetrization. Secondly, for these convex bodies, a stability version of the characterization of ellipsoids by the fact that each hyperplane section is centrally symmetric is established.

Comparison of volumes of convex bodies in real, complex, and quaternionic spaces

The classical Busemann–Petty problem (1956) asks, whether origin-symmetric convex bodies in R n with smaller hyperplane central sections necessarily have smaller volumes. It is known, that the answer is affirmative if n ⩽ 4 and negative if n > 4 . The same question can be asked when volumes of hyperplane sections are replaced by other comparison functions having geometric meaning. We give unified analysis of this circle of problems in real, complex, and quaternionic n-dimensional spaces. All cases are treated simultaneously. In particular, we show that the Busemann–Petty problem in the quaternionic n-dimensional space has an affirmative answer if and only if n = 2 . The method relies on the properties of cosine transforms on the unit sphere. We discuss possible generalizations.

Primitive cohomology and the tube mapping

Let X be a smooth complex projective variety of dimension d. We show that its primitive cohomology in degree d is generated by certain “tube classes,” constructed from the monodromy in the family of all hyperplane sections of X. The proof makes use of a result about the group cohomology of certain representations that may be of independent interest.

On the local Picard group

In his book SGA2, A. Grothendieck developed Lefschetz theorems for the Picard group, the aim being to compare the Picard group of a projective variety with the one of a hyperplane section. An intermediate object is the Picard group of the formal completion along the hyperplane section. Here we proceed similarly but in the local complex analytic context. The use of the exponential sequence leads to analytic as well as topological depth conditions.

Fonction asymptotique de Samuel des sections hyperplanes et multiplicité

Let ( A , m A , k ) be a local noetherian ring and I an m A -primary ideal. The asymptotic Samuel function (with respect to I ) v I ¯ : A ⟶ R ∪ { + ∞ } is defined by v I ¯ ( x ) = l i m k → + ∞ o r d I ( x k ) k , ∀ x ∈ A . Similarly, one defines, for another ideal J , v I ¯ ( J ) as the minimum of v I ¯ ( x ) as x varies in J . Of special interest is the rational number v I ¯ ( m A ) . We study the behavior of the asymptotic Samuel function (with respect to I ) when passing to hyperplane sections of A as one does for the theory of mixed multiplicities.

Veronesean embeddings of Hermitian unitals

In this paper, we determine the Veronesean embeddings of Hermitian unitals, i.e., the representations of Hermitian unitals as points of a 7-dimensional projective space where the blocks are plane ovals. As an application, we derive that the following objects coincide: (1) the generic hyperplane sections of Hermitian Veroneseans in an 8-dimensional projective space, (2) the Grassmannians of the classical spreads of non-degenerate quadrics of Witt index 2 in a 5-dimensional projective space, (3) the sets of absolute points of trialities of Witt index 1. As a consequence, we prove that the set of absolute points of a triality without fixed lines, but with absolute points, determines the triality quadric and the triality itself uniquely.

Stable determination of convex bodies from sections

A star body (with respect to the origin 0) in ℝ d ( d ≧ 3) which has 0 as center of symmetry is uniquely determined by the ( d − 1)-dimensional volumes of its sections with hyperplanes through 0. Without the symmetry assumption, we show that a star body is uniquely determined by the volumes and centroids of its hyperplane sections through 0. For convex bodies, we prove a stability version of this result.

ELLIPTIC CURVES RELATED TO CYCLIC CUBIC EXTENSIONS

The aim of this paper is to study certain family of elliptic curves [Formula: see text] defined over a number field F arising from hyperplane sections of some cubic surface [Formula: see text] associated to a cyclic cubic extension K/F. We show that each [Formula: see text] admits a 3-isogeny ϕ over F and the dual Selmer group [Formula: see text] is bounded by a kind of unit/class groups attached to K/F. This is proven via certain rational function on the elliptic curve [Formula: see text] with nice property. We also prove that the Shafarevich–Tate group [Formula: see text] coincides with a class group of K as a special case.

The structure of algebraic embeddings of $\mathbb{C}^{2}$ into $\mathbb{C}^{3}$ (the normal quartic hypersurface case. II)

We obtain the affirmative answer for a special case of the linearization problem for algebraic embeddings of $\mathbb{C}^{2}$ into $\mathbb{C}^{3}$. Indeed, we determine all the compactifications $(X,Y)$ of $\mathbb{C}^{2}$ such that $X$ are normal quartic hypersurfaces in $\mathbb{P}^{3}$ without triple points and $Y$ are hyperplane sections of $X$. Moreover, for each $(X,Y)$, we construct a tame automorphism of $\mathbb{C}^{3}$ which transforms the hypersurface $X\setminus Y$ onto a coordinate hyperplane.