Smooth Hypersurface Of Degree Research Articles

K-stability of Casagrande–Druel varieties

Abstract We introduce a new subclass of Fano varieties (Casagrande–Druel varieties) that are 𝑛-dimensional varieties constructed from Fano double covers of dimension n − 1 n-1 . We conjecture that a Casagrande–Druel variety is K-polystable if the double cover and its base space are K-polystable. We prove this for smoothable Casagrande–Druel threefolds, and for Casagrande–Druel varieties constructed from double covers of P n − 1 \mathbb{P}^{n-1} ramified over smooth hypersurfaces of degree 2 ⁢ d 2d with n > d > n 2 > 1 n>d>\frac{n}{2}>1 . As an application, we describe the connected components of the K-moduli space parametrizing smoothable K-polystable Fano threefolds in the families № 3.9 and № 4.2 in the Mori–Mukai classification.

The discriminant of a hypersurface in weighted projective space

In this paper, we study the discriminant of a hypersurface in a weighted projective space with isolated singularities. We define the discriminant of a hypersurface in [Formula: see text] with [Formula: see text], and show that it satisfies a smoothness criterion over any commutative ring. A smooth hypersurface of degree [Formula: see text] in the weighted projective three-fold [Formula: see text] is a del Pezzo surface of degree [Formula: see text]. We show that the determinant of the Galois action on the [Formula: see text]-lattice arising from its exceptional curves can be computed via the square roots of the discriminant.

On initialized and ACM line bundles over a smooth sextic surface in

Let be a smooth hypersurface of degree 6 over complex numbers. In this article, we give a characterization of initialized and ACM bundles of rank 1 on X with respect to the line bundle given by a smooth hyperplane section of X.

Stability of the tangent bundles of complete intersections and effective restriction

For n≥3 and r≥1, let M be an (n+r)-dimensional irreducible Hermitian symmetric space of compact type and let 𝒪 M (1) be the ample generator of pic(M). Let Y=H 1 ∩⋯∩H r be a smooth complete intersection of dimension n, where H i ∈|𝒪 M (d i )| with d i ≥2. We prove a vanishing theorem for twisted holomorphic forms on Y. As an application, we show that the tangent bundle T Y of Y is stable. Moreover, if X is a smooth hypersurface of degree d in Y such that the restriction pic(Y)→pic(X) is surjective, we establish some effective results for d to guarantee the stability of the restriction T Y | X . In particular, if Y is a general hypersurface in ℙ n+1 and X is a general smooth divisor in Y, we show that T Y | X is stable except for some well-known examples. We also address the cases where the Picard group increases by restriction.

On abelian automorphism groups of hypersurfaces

Given integers d ≥ 3 and N ≥ 3, let G be a finite abelian group acting faithfully and linearly on a smooth hypersurface of degree d in the complex projective space ℙN−1. Suppose G ⊂ PGL(N, ℂ) can be lifted to a subgroup of GL(N, ℂ). Suppose moreover that there exists an element g in G such that G/❬g❭ has order coprime to d − 1. Then all possible G are determined (Theorem 4.3). As an application, we derive (Theorem 4.8) all possible orders of linear automorphisms of smooth hypersurfaces for any given (d, N). In particular, we show (Proposition 5.1) that the order of an automorphism of a smooth cubic fourfold is a factor of 21, 30, 32, 33, 36 or 48, and each of those 6 numbers is achieved by a unique (up to isomorphism) cubic fourfold.

Kodaira dimension of moduli of special cubic fourfolds

Abstract A special cubic fourfold is a smooth hypersurface of degree 3 and dimension 4 that contains a surface not homologous to a complete intersection. Special cubic fourfolds give rise to a countable family of Noether–Lefschetz divisors 𝒞 d {{\mathcal{C}}_{d}} in the moduli space 𝒞 {{\mathcal{C}}} of smooth cubic fourfolds. These divisors are irreducible 19-dimensional varieties birational to certain orthogonal modular varieties. We use the “low-weight cusp form trick” of Gritsenko, Hulek, and Sankaran to obtain information about the Kodaira dimension of 𝒞 d {{\mathcal{C}}_{d}} . For example, if d = 6 ⁢ n + 2 {d=6n+2} , then we show that 𝒞 d {{\mathcal{C}}_{d}} is of general type for n > 18 {n>18} , n ∉ { 20 , 21 , 25 } {n\notin\{20,21,25\}} ; it has nonnegative Kodaira dimension if n > 13 {n>13} and n ≠ 15 {n\neq 15} . In combination with prior work of Hassett, Lai, and Nuer, our investigation leaves only twenty values of d for which no information on the Kodaira dimension of 𝒞 d {{\mathcal{C}}_{d}} is known. We discuss some questions pertaining to the arithmetic of K3 surfaces raised by our results.

The relative Hilbert scheme of projection morphisms

Let [Formula: see text] be a smooth hypersurface of degree [Formula: see text] in a projective space [Formula: see text] and take a point [Formula: see text] in [Formula: see text]. Let [Formula: see text] be the relative Hilbert scheme parametrizing zero-dimensional subscheme, of length [Formula: see text], of fibers of the projection morphism [Formula: see text] from [Formula: see text]. In this paper we present an embedding of the relative Hilbert scheme [Formula: see text] into a weighted projective space and describe its defining ideal for general [Formula: see text]. We also study line bundles on the relative Hilbert scheme [Formula: see text] for [Formula: see text] and general [Formula: see text].

Varieties of minimal rational tangents on double covers of projective space

Let \(\phi : X \rightarrow \mathbb{P }^n\) be a double cover branched along a smooth hypersurface of degree \(2m, 2 \le m \le n-1\). We study the varieties of minimal rational tangents \(\mathcal{C }_x \subset \mathbb{P }T_x(X)\) at a general point \(x\) of \(X\). We describe the homogeneous ideal of \(\mathcal{C }_x\) and show that the projective isomorphism type of \(\mathcal{C }_x\) varies in a maximal way as \(x\) varies over general points of \(X\). Our description of the ideal of \(\mathcal{C }_x\) implies a certain rigidity property of the covering morphism \(\phi \). As an application of this rigidity, we show that any finite morphism between such double covers with \(m=n-1\) must be an isomorphism. We also prove that Liouville-type extension property holds with respect to minimal rational curves on \(X\).

Birationally rigid hypersurfaces

We prove that for N≥4, all smooth hypersurfaces of degree N in ℙ N are birationally superrigid. First discovered in the case N=4 by Iskovskikh and Manin in a work that started this whole direction of research, this property was later conjectured to hold in general by Pukhlikov. The proof relies on the method of maximal singularities in combination with a formula on restrictions of multiplier ideals.

Nonuniruledness results for spaces of rational curves in hypersurfaces

We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree d in P are not uniruled if (n + 1)/2 ≤ d ≤ n − 3. We also show that for any e ≥ 1, the space of smooth rational curves of degree e in a general hypersurface of degree d in P is not uniruled roughly when d ≥ e √ n.

Restriction of the Tangent Bundle of G/P to a Hypersurface

AbstractLet P be a maximal proper parabolic subgroup of a connected simple linear algebraic group G, defined over ℂ, such that n := dimℂG/P ≥ 4. Let ι : Z ↪ G/P be a reduced smooth hypersurface of degree at least (n – 1) · degree(T(G/P))/n. We prove that the restriction of the tangent bundle ι*TG/P is semistable.

Arithmetically Cohen–Macaulay bundles on complete intersection varieties of sufficiently high multidegree

Recently it has been proved that any arithmetically Cohen-Macaulay (ACM) bundle of rank two on a general, smooth hypersurface of degree at least three and dimension at least four is a sum of line bundles. When the dimension of the hypersurface is three, a similar result is true provided the degree of the hypersurface is at least six. We extend these results to complete intersection subvarieties by proving that any ACM bundle of rank two on a general, smooth complete intersection subvariety of sufficiently high multi-degree and dimension at least four splits. We also obtain partial results in the case of threefolds.

Equations différentielles sur les hypersurfaces de [formula omitted

In this paper we prove that every entire curve in a smooth hypersurface of degree d ⩾ 97 in P C 4 must satisfy an algebraic differential equation of order 3. The logarithmic version of this result is given proving that every entire curve in the complement of a smooth surface of degree d ⩾ 92 in P C 3 must satisfy an algebraic differential equation of order 3.

Double cubics and double quartics

We study a double cover branched over a smooth divisor such that R is cut on V by a hypersurface of degree 2(n−deg(V)), where n ≥ 8 and V is a smooth hypersurface of degree 3 or 4. We prove that X is nonrational and birationally superrigid.

Factoriality of Certain Threefolds Complete Intersection in P 5 with Ordinary Double Points#

Let V ⊂ P 5 be a reduced and irreducible threefold of degree s, complete intersection on a smooth hypersurface of degree t, with s > t 2 − t. In this paper, we prove that if the singular locus of V consists of δ < 3s/8t ordinary double points, then any projective surface contained in V is a complete intersection on V. In particular, V is Q-factorial.

Total log canonical thresholds and generalized Eckardt points

Let be a smooth hypersurface of degree in . It is proved that the log canonical threshold of an arbitrary hyperplane section of it is at least . Under the assumption of the log minimal model program it is also proved that the log canonical threshold of is if and only if is a cone in over a smooth hypersurface of degree in .

Hyperplane sections of Calabi-Yau varieties

Theorem: If W is a smooth complex projective variety with h^1 (O-script_W) = 0, then a sufficiently ample smooth divisor X on W cannot be a hyperplane section of a Calabi-Yau variety, unless W is itself a Calabi-Yau. Corollary: A smooth hypersurface of degree d in P^n (n >= 2) is a hyperplane section of a Calabi-Yau variety iff n+2 <= d <= 2n+2. The method is to construct out of the variety W a universal family of all varieties Z for which X is a hyperplane section with normal bundle K_X, and examine the bad singularities of such Z. A motivation is to show many curves cannot be divisors on a K-3 surface.

Composantes de dimension maximale d'un analogue du lieu de Noether-Lefschetz (Components of Maximal Dimension of an Analogue of the Noether-Lefschetz Locus)

Let X [subset or is implied by] $\mathbb P$4$\mathbb _C$ be a smooth hypersurface of degree d [gt-or-equal, slanted] 5, and let S [subset or is implied by] X be a smooth hyperplane section. Assume that there exists a non trivial cycle Z [set membership] Pic(X) of degree 0, whose image in CH1(X) is in the kernel of the Abel–Jacobi map. The family of couples (X, S) containing such Z is a countable union of analytic varieties. We show that it has a unique component of maximal dimension, which is exaclty the locus of couples (X, S) satisfying the following condition: There exists a line Δ [subset or is implied by] S and a plane P [subset or is implied by] $\mathbb P$4$_{\mathbb C}$ such that P [cap B: intersection] X = Δ, and Z = Δ − dh, where h is the class of the hyperplane section in CH1(S). The image of Z in CH1(X) is thus 0. This construction provides evidence for a conjecture by Nori which predicts that the Abel–Jacobi map for 1–cycles on X is injective.

Chernklassen semi-stabiler Rang-3-vektorb�ndel auf kubischen Dreimannigfaltigkeiten

It is shown that the third Chern-number of a semistable Rk-3-vector bundle on a smooth hypersurface of degree 3 in ℙ4 can be bounded by the first and the second Chern-number of the bundle.

THREE-DIMENSIONAL QUARTICS AND COUNTEREXAMPLES TO THE LÜROTH PROBLEM

In this paper we prove that any birational mapping between smooth hypersurfaces of degree four is an isomorphism. Since B. Segre constructed examples of smooth unirational quartics, this leads to a negative resolution of the three-dimensional Luroth problem. Bibliography: 13 items.