Related papers: Degrees of closed points on hypersurfaces

For a closed minimal immersed hypersurface $M$ in $\mathbb S^{n+1}$ with second fundamental form $A$, and each integer $k\ge 2$, define a constant $\sigma_k=\dfrac{\int_M (|A|^2)^k}{|M|}$. We show that $\sigma_k \ge 2^k$ provided $n=2$ and…

Explicit formulas determining the dimension and the degree of the singular subscheme of hypersurfaces in ${\mathbb P}^n$ are given in terms of the graded Betti numbers of the minimal free resolution of the corresponding Jacobian algebra.…

The $k$-flex locus of a projective hypersurface $V\subset \mathbb P^n$ is the locus of points $p\in V$ such that there is a line with order of contact at least $k$ with $V$ at $p$. Unexpected contact orders occur when $k\ge n+1$. The case…

We prove an effective bound for the degree of a smooth divisor of a hypersurface of P^n, n>4 (projective space over an algebraically closed field of characteristic zero). Our result follows from a strong (since the degree of the divisor is…

A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes passing below…

Let $r \geq 2$ be a fixed integer. For infinitely many $n$, let $\boldsymbol{k} = (k_1,..., k_n)$ be a vector of nonnegative integers such that their sum $M$ is divisible by $r$. We present an asymptotic enumeration formula for simple…

We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree $d$ in $P^n$ are not uniruled if $(n+1)/2 \leq d \leq n-3$. We also show that for any positive integer $e$, the space of smooth…

If $(A,\mathfrak{m})$ is a hypersurface ring of dimension $d$ with $e(A)=3$. Let $M$ be an MCM $A$-module with $\mu(M)=4$ then we prove that $\depth{G(M)}\geq d-3$.

We study the set of $D$ such that a given irreducible hypersurface $C$ of degree $d$ has infinitely many points of degree $D$ over $\mathbb{Q}$. We give a new explicit proof that this set contains all (positive) multiples of the index of…

Let k be a complete, non-Archimedean field and let X be a k-analytic space ; assume that there exists a tamely ramified finite extension L/k such that X_L is isomorphic to an open polydisc over L ; we prove that X is itself isomorphic to an…

We prove that for any compact quasi-smooth strictly $k$-analytic space $X$ there exist a finite extension $l/k$ and a quasi-\'etale covering $X'\to X\otimes_kl$ such that $X'$ possesses a strictly semistable formal model. This extends a…

In Dokchitser (2007) it is shown that given an elliptic curve $E$ defined over a number field $K$ then there are infinitely many degree 3 extensions $L/K$ for which the rank of $E(L)$ is larger than $E(K)$. In the present paper we show that…

Let $K$ be a discretely valued Henselian field. Creutz and Viray show that the degree set of a curve $C$ over a $p$-adic field can miss infinitely many multiples of the index of $C$, a phenomenon that cannot occur over finitely generated…

In the study of immersed surfaces of constant positive extrinsic curvature in space-forms, it is natural to substitute completeness for a weaker property, which we here call quasicompleteness. We determine the global geometry of such…

We classify linearly normal surfaces $S \subset \mathbf{P}^{r+1}$ of degree $d$ such that $4g-4 \leq d \leq 4g+4$, where $g>1$ is the sectional genus (it is a classical result that for larger $d$ there are only cones). We apply this to the…

Given a K3 surface X over a field of characteristic p, Artin conjectured that if X is supersingular (meaning infinite height) then its Picard rank is 22. Along with work of Nygaard-Ogus, this conjecture implies the Tate conjecture for K3…

Let $A$ be an abelian scheme of dimension at least four over a $\mathbb{Z}$-finitely generated integral domain $R$ of characteristic zero, and let $L$ be an ample line bundle on $A$. We prove that the set of smooth hypersurfaces $D$ in $A$…

In this paper, we consider Weingarten curvature equations for $k$-convex hypersurfaces with $n<2k$ in a warped product manifold $\overline{M}=I\times_{\lambda}M$. Based on the conjecture proposed by Ren-Wang in \cite{Ren2}, which is valid…

Let $K = K(C)$ be the function field of a smooth curve $C$. Applying the result of [Xu08], we prove that if $S/K$ is a degree one or two del Pezzo surface which can be completed to a generic family in the parametrizing space over $C$, then…

We construct nontrivial L-equivalence between curves of genus one and degree five, and between elliptic surfaces of multisection index five. These results give the first examples of L-equivalence for curves (necessarily over…