The aim of these lectures is to study rational points and rational curves on varieties, mainly over finite fields Fq. We concentrate on hypersurfaces Xn of degree ≤ n+ 1 in Pn+1, especially on cubic hypersurfaces. The theorem of Chevalley–Warning (cf. Esnault’s lectures) guarantees rational points on low degree hypersurfaces over finite fields. That is, if X ⊂ Pn+1 is a hypersurface of degree ≤...

We consider Potts model hypersurfaces defined by the multivariate Tutte polynomial of graphs (Potts model partition function). We focus on the behavior of the number of points over finite fields for these hypersurfaces, in comparison with the graph hypersurfaces of perturbative quantum field theory defined by the Kirchhoff graph polynomial. We give a very simple example of the failure of the “f...

In this talk, first we introduce the classification of homogeneous hypersurfaces in some Hermitian symmetric spaces of rank 1 or rank 2. In particular, we give a full expression of the geometric structures for hypersurfaces in complex two-plane Grassmannians G2(C) or in complex hyperbolic twoplane Grassmannians G2(C). Next by using the isometric Reeb flow we give a complete classification for h...

In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with P + aH = b in a locally symmetric Lorentz space Ln+1 1 . Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces Ln+1 1 satisfying some curvature conditions. By modifying Cheng-Yau’s operator given in [7], we introduce a modified operator L and...

In this paper we describe all rotation H-hypersurfaces in Hn × R and use them as barriers to prove existence and characterization of certain vertical H-graphs and to give symmetry and uniqueness results for compact H-hypersurfaces whose boundary is one or two parallel submanifolds in slices. We also describe examples of translation H-hypersurfaces in Hn × R. For n > 2 we obtain a complete embed...

The first part of this paper gives a complete description of local automorphism groups for Levi degenerate hypersurfaces of finite type in C. We also prove that, with the exception of hypersurfaces of the form v = |z|, local automorphisms are always determined by their 1-jets. Using this result, in the second part we describe special normal forms which by an additional normalization eliminate t...

This paper studies hypersurfaces admitting a locally symmetric connection which is induced by the Gauß conormal map in affine geometry. It is known that the rank of the shape operator is of importance to this topic. In dimension two we give new results for an arbitrary shape operator. In the case of a nondegenerate shape operator hypersurfaces with locally symmetric conormal connection can be t...

The constraints on the asymptotic behavior of the conformal factor and conformal extrinsic curvature imposed by the initial value equations of general relativity on constant mean extrinsic curvature (CMC) hypersurfaces are analyzed in detail. We derive explicit formulas for the Bondi-Sachs energy and momentum in terms of coefficients of asymptotic expansions on CMC hypersurfaces near future nul...

Theorem. There exist no C Levi-flat real hypersurfaces in CP . This improves an earlier result of Siu [Si2] where C smoothness is required. For the nonexistence of Levi-flat hypersurfaces in CP with n ≥ 3, Lins-Neto [LN] first proved the nonexistence of real-analytic hypersurfaces in CP. Nonexistence of C Levi-flat hypersurfaces in CP was proved for n ≥ 3 by Siu [Si1]. It is proved in a recent ...

We study singular real-analytic Levi-flat hypersurfaces in complex projective space. We give necessary and sufficient conditions for such a hypersurface to be a pullback of a real-analytic curve in C via a meromorphic function. We define the rank of a real hypersurface and study the connections between rank, degree, and the type and size of the singularity for Levi-flat hypersurfaces. Finally, ...