Matrix weighted rational curves and surfaces sciencedirect. Rational curves of degree at most 9 on a general quintic threefold. Counting rational points on hypersurfaces of low dimension. Chapter 2 hypersurfaces in weighted projective spaces over. Thus we get the inequality j 2kj qdeg, for each k 1. Counting rational curves on k3 surfaces arnaud beauville 1 introduction the aim of these notes is to explain the remarkable formula found by yau and zaslow. In this paper, we prove that, if an embedded smooth projective manifold is swept out by high dimensional hypersurfaces of degree d, then either it is a scroll, or it admits an extremal contraction whose general fibers are hypersurfaces of degree d, under the assumption of hartshornes famous conjecture on complete intersections. Given a hypersurface x, let r ex be the space of integral, geometrically rational curves of degree eon x. We end with a discussion of ongoing research on the moduli space of rational curves of xed degree on a general hypersurface.
Since n e does not intersect the locus of smooth rational curves of degree e, it follows that the locus of smooth rational curves must be a zariski open set in the other irreducible component. This is a continuation of rational curves on hypersurfaces of low degree, math. Nonrationality of hypersurfaces summer term 2015, tuesday 24pm, 0. Citeseerx rational curves on hypersurfaces of low degree. The following plan is tentative, and open for discussion. For example, cayley wrote in a 1869 memoir that a smooth complex cubic d 3 surface contains exactly 27 projective lines. It also has implications about the hilbert functions of rational curves of low degree in pn. The incidence correspondences will not be irreducible in general, so this result is an exploration of when things work out as well as possible. As an application, we prove that the space of lines on any smooth fano hypersurface of. We prove a conjecture of voisin that no two distinct points on a very genera.
Journal of singularities volume 9 2014, 219225 proc. By popular request, we will focus more on hodge theory rather than derived categories. Integral curves in null hypersurfaces stack exchange. Hypersurfaces are a natural rst case for these sorts of questions, and much work has been done investigating the dimensions of spaces of rational curves on hypersurfaces. Although it becomes more dicult in the case n\geq 4. The sets of zeroes in the projective space of homogeneous polynomials of degree dwith coe cients in a eld are called projective hypersurfaces of degree dand the study of their geometry is a very classical subject. Convex hypersurfaces with pinched principal curvatures and. The main technical issues revolve around the existence of free rational curves with prescribed geometric properties.
For a general hypersurface of degree din projective nspace, if n. Deformation of singularities and the homology of intersection spaces markus banagl and laurentiu maxim abstract. At0502506 v2 15 oct 2005 boundary manifolds of projective hypersurfaces daniel c. Our method provides systematic constructions of a the beauville. Decent intersection and torrigidity for modules over. Introduction in this survey, we will describe some results on families of rational curves on hypersurfaces in pn.
We have seen that being unirational is a weaker condition on x than being rational. Rigidity of hypersurfaces j anos koll ar princeton university. On manifolds swept out by high dimensional hypersurfaces. Osullivan, and samrith ram abstract we consider the question of determining the maximum number of f qrational points that can lie on a hypersurface of a given degree in. Skip to main content this banner text can have markup. Counting rational curves on k3 surfaces universiteit leiden. Rational curves on hypersurfaces of low degree, ii. Indeed, through a general point of such hypersurface f, there is exactly.
Rm,n, known to imply decent intersection, also implies rigidity. Letnx,b be the number of rational points of height at most b on a variety x. A 3dimensional curve is a curve of general type if its first two derivatives are not parallel. But if 2is an eigenvalue of frobjf, then k is an eigenvalue of frobjf2k. We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in lorentzian manifolds, where one has a lower bound.
Then every birational map x 99k x0 to any fano variety x0is an isomorphism. Our argument will use the \explicit nature of hypersurfaces, so it does not generalize. Louis washington university open scholarship all theses and dissertations etds 5242012 castelnuovomumford regularity of general rational curves o. A note on gonality of curves on general hypersurfaces. Hypersurfaces of low entropy and mean curvature ow lu wang in this talk, i will present joint work with jacob bernstein on properties of hypersurfaces of low entropy using mean curvature ow mcf. Hypersurfaces with too many rational curves springerlink. Remarkswe have three di erent proofs of this existence theorem i bogomolovmumford. Weak approximation for hypersurfaces of low degree brendan hassett and yuri tschinkel abstract. Hypersurfaces of low degree are rationally simplyconnected jason michael starr abstract. While intersection cohomology is stable under small resolutions, both ordinary and intersection cohomology are unstable under smooth deformation of singularities. Introduction the geometry of a variety is intimately linked to the geometry of the space of rational curves on it. From now on we shall suppose that the curves in question are all of general type. For complex projective algebraic hypersurfaces with an isolated. Rational curves on general hypersurfaces of degree.
Starr and harris 10 show that for d general degree d hypersurface contains the. Multiple solutions of a critical polyharmonic equation. In this section we present the definition of general matrix weighted rational curves and surfaces. If we denote by r ex, the family of all smooth rational curves of degree ewith respect to a given polarization. March 11, 2014 although a projective k3 surface over the complex numbers cannot be unirational, bogomolov conjectured that it contains nevertheless. There is a natural conjecture, made in special cases by several di erent people, for the dimension of r e x for xvery general. The case of families of rational curves on pn itself is well understood. In hilbert s case, we can determine these hyperplane sections by intuition.
Hypersurfaces in weighted projective spaces over finite fields with applications to coding theory yves aubry, wouter castryck, sudhir r. The distinguished frenet frame vector fields t, t and t of a 1 2 3. We explore the arithmetic of rationally connected varieties over function. Plucker coordinates as a surface of degree 10 and contains smooth rational curves of degree less or equal to 4.
A diamagnetic inequality for semigroup differences. Seminar on cubic hypersurfaces tentative plan ws 201617 the goal for this reading seminar is to learn something about cubic threefolds and fourfolds, with an emphasis on the rationality questions. Consider a hypersurface of degree 42 in the weighted projective space p 6,14,21,1. Convex hypersurfaces with pinched principal curvatures and flow of convex hypersurfaces by high powers of curvature abstract we consider convex hypersurfaces for which the ratio of principal curvatures at each point is bounded by a function of the maximum principal curvature with limit 1 at infinity. Special numbers of rational points on hypersurfaces in the dimensional projective space over a finite field.
On the geometry of hypersurfaces of low degrees in the. Higher order curvature flow of curves and hypersurfaces. Special numbers of rational points on hypersurfaces in the. A method for defining the weight matrices using specified normals and some basic properties of the obtained curves and. We consider contraction of convex hypersurfaces by convex speeds, homogeneous of degree one in the principal curvatures, that are not necessarily smooth. There has also been progress in the area of rational curves lying on hypersurfaces of low degree. Results on density of rational curves over the standard topology have recently been obtain by chen and lewis cl10. Basics of matrix weighted rational curves and surfaces. Bulletin of the malaysian mathematical sciences society, vol. Curvature contraction of convex hypersurfaces by nonsmooth. Classical results, going back to clemens, gri ths, manin et al, show e.
The contribution from the minor arcs 15 references 18 1. Download citation rational curves on hypersurfaces of low degree, ii this is the second in a sequence of papers on the geometry of spaces of rational curves of degree e on a general. We study rational points and rational curves on varieties over. We show how to approximate such a speed by a sequence of smooth speeds for which behaviour is well known.