General Hypersurface Of Degree Research Articles

Interpolation of curves on Fano hypersurfaces

On a general hypersurface of degree [Formula: see text] in [Formula: see text] or [Formula: see text] itself, we prove the existence of curves of any genus and high enough degree depending on the genus passing through the expected number [Formula: see text] of general points or incident to a general collection of subvarieties of suitable codimensions. In some cases, we also show that the family of curves through [Formula: see text] fixed points has general moduli as family of [Formula: see text]-pointed curves. These results imply positivity of certain intersection numbers on Kontsevich spaces of stable maps. An arithmetical Appendix A by Chang describes the set of numerical characters ([Formula: see text], curve degree, genus) to which our results apply.

Cones of lines having high contact with general hypersurfaces and applications

AbstractGiven a smooth hypersurface of degree , we study the cones swept out by lines having contact order at a point . In particular, we prove that if X is general, then for any and , the cone has dimension exactly . Moreover, when X is a very general hypersurface of degree , we describe the relation between the cones and the degree of irrationality of k‐dimensional subvarieties of X passing through a general point of X. As an application, we give some bounds on the least degree of irrationality of k‐dimensional subvarieties of X passing through a general point of X, and we prove that the connecting gonality of X satisfies .

Clustered families and applications to Lang‐type conjectures

We introduce and classify 1-clustered families of linear spaces in the Grassmannian G ( k − 1 , n ) $\mathbb {G}(k-1,n)$ and give applications to Lang-type conjectures. Let X ⊂ P n $X \subset \mathbb {P}^n$ be a very general hypersurface of degree d $d$ . Let Z L $Z_L$ be the locus of points contained in a line of X $X$ . Let Z 2 $Z_2$ be the closure of the locus of points on X $X$ that are swept out by lines that meet X $X$ in at most 2 points. We prove that: if d ⩾ 3 n + 2 2 $d \geqslant \frac{3n+2}{2}$ , then X $X$ is algebraically hyperbolic modulo Z L ; $Z_L\hbox{\it ;}$ if d ⩾ 3 n 2 $d \geqslant \frac{3n}{2}$ , X $X$ contains lines but no other rational curves; if d ⩾ 3 n + 3 2 $d \geqslant \frac{3n+3}{2}$ , then the only points on X $X$ that are rationally Chow zero equivalent to points other than themselves are contained in Z 2 $Z_2$ ; if d ⩾ 3 n + 2 2 $d \geqslant \frac{3n+2}{2}$ and a relative Green–Griffiths–Lang Conjecture holds, then the exceptional locus for X $X$ is contained in Z 2 $Z_2$ . These sharpen prior results of Ein, Voisin, Pacienza, Clemens and Ran.

Stable rationality of index one Fano hypersurfaces containing a linear space

We prove that a very general complex hypersurface of degree \(n+1\) in \(\mathbb {P}^{\,n+1}\) containing an r-plane with multiplicity m is not stably rational under some mild assumptions for \(n \geqslant 3\), \(m, r > 0\). We also prove the failure of stable rationality of a very general hypersurface of degree \(n+1\) in \(\mathbb {P}^{\,n+1}\) admitting several isolated ordinary double points.

Rational curves on general type hypersurfaces

We develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of Clemens and Ran to prove that a very general hypersurface of degree $\frac{3n+1}{2} \leq d \leq 2n - 3$ in $\mathbb{P}^n$ contain lines but no other rational curves.

Positivity results for spaces of rational curves

Let $X$ be a very general hypersurface of degree $d$ in $\mathbb{P}^n$. We investigate positivity properties of the spaces $R_e(X)$ of degree $e$ rational curves in $X$. We show that for small $e$, $R_e(X)$ has no rational curves meeting the locus of smooth embedded curves. We show that for $n \leq d$, there are no rational curves in the locus $Y \subset X$ swept out by lines. And we exhibit differential forms on a smooth compactification of $R_e(X)$ for every $e$ and $n-2 \geq d \geq \frac{n+1}{2}$.

Gonality of curves on general hypersurfaces

This paper concerns the existence of curves with low gonality on smooth hypersurfaces of sufficiently large degree. It has been recently proved that if X⊂Pn+1 is a hypersurface of degree d⩾n+2, and if C⊂X is an irreducible curve passing through a general point of X, then its gonality verifies gon(C)⩾d−n, and equality is attained on some special hypersurfaces. We prove that if X⊂Pn+1 is a very general hypersurface of degree d⩾2n+2, the least gonality of an irreducible curve C⊂X passing through a general point of X is gon(C)=d−⌊16n+1−12⌋, apart from a series of possible exceptions, where gon(C) may drop by one.

A note on gonality of curves on general hypersurfaces

This short paper concerns the existence of curves with low gonality on smooth hypersurfaces $$X\subset \mathbb {P}^{n+1}$$ . After reviewing a series of results on this topic, we report on a recent progress we achieved as a product of the Workshop Birational geometry of surfaces, held at University of Rome “Tor Vergata” on January 11th–15th, 2016. In particular, we obtained that if $$X\subset \mathbb {P}^{n+1}$$ is a very general hypersurface of degree $$d\geqslant 2n+2$$ , the least gonality of a curve $$C\subset X$$ passing through a general point of X is $$\mathrm {gon}(C)=d-\left\lfloor \frac{\sqrt{16n+1}-1}{2}\right\rfloor $$ , apart from some exceptions we list.

Cyclic covers that are not stably rational

Using methods developed by Kollár, Voisin, ourselves and Totaro, we prove that a cyclic cover of , , of prime degree , ramified along a very general hypersurface of degree , is not stably rational if . In dimension 3 we recover double covers of ramified along a very general surface of degree 4 (Voisin) and double covers of ramified along a very general surface of degree 6 (Beauville). We also find double covers of ramified along a very general hypersurface of degree 6. This method also enables us to produce examples over a number field.

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.

Kobayashi hyperbolic imbeddings into toric varieties

Our main goal of this article is to give a characterization of an algebraic divisor on an algebraic torus whose complement is Kobayashi hyperbolically imbedded into a toric projective variety. As an application of our main theorem, we prove the following: the complement of the union of n + 1 hyperplanes in the n-dimensional projective space \({\mathbb{P}^{n}(\mathbb{C})}\) in general position and a general hypersurface of degree n in \({\mathbb{P}^n(\mathbb{C})}\) is Kobayashi hyperbolically imbedded into \({\mathbb{P}^n(\mathbb{C})}\).

Arithmetically Cohen–Macaulay bundles on hypersurfaces

We prove that any rank two arithmetically Cohen–Macaulay vector bundle on a general hypersurface of degree at least three in ℙ^5 must be split.

On the logarithmic Kobayashi conjecture

We study the hyperbolicity of the log variety (ℙn,X), where X is a very general hypersurface of degree d ≧ 2n + 1 (which is the bound predicted by the Kobayashi conjecture). Using a positivity result for the sheaf of (twisted) logarithmic vector fields, which may be of independent interest, we show that any log-subvariety of (ℙn, X) is of log-general type, give a new proof of the algebraic hyperbolicity of (ℙn, X), and exclude the existence of maximal rank families of entire curves in the complement of the universal degree d hypersurface. Moreover, we prove that, as in the compact case, the algebraic hyperbolicity of a log-variety is a necessary condition for the metric one.

Arithmetically Cohen-Macaulay Bundles on Three Dimensisonal Hypersurfaces

We prove that any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least six in ℙ4 must be split

Birational geometry of Fano direct products

We prove the birational superrigidity of direct products of primitive Fano varieties of the following two types: either is a general hypersurface of degree , , or is a general double space of index 1, . In particular, every structure of a rationally connected fibre space on is given by the projection onto a direct factor. The proof is based on the connectedness principle of Shokurov and Kollár and the technique of hypertangent divisors.

Curves on threefolds with trivial canonical bundle

C H Clemens has shown that homologically trivial codimension two cycles on a general hypersurface of degree five and dimension three form a subgroup of infinite rank inside the intermediate jacobian. We generalize this to other complete intersection threefolds with trivial canonical bundle.

Elliptic curves on the general hypersurface of degree 7 of ℙ 4

In this paper we prove that, if C is an elliptic curve of degree d on a general hypersurface V of degree 7 of ℙ 4 , then d is a multiple of 7.