Related papers: Linear subspaces of minimal codimension in hypersu…

Our main theorem characterizes the complete intersections of codimension 2 in a projective space of dimension 3 or more over an algebraically closed field of characteristic 0 as the subcanonical and self-linked subschemes. In order to prove…

Let $n$ and $p$ be non-negative integers with $n \geq p$, and $S$ be a linear subspace of the space of all $n$ by $p$ matrices with entries in a field $\mathbb{K}$. A classical theorem of Flanders states that $S$ contains a matrix with rank…

We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let $X$ be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension $d$. Informally, the theorem states that if $X$…

Let $k$ be a field and $V$ an $k$-vector space. For a family $\bar P=\{ P_i\}_{1\leq i\leq c}, $ of polynomials on $V$, we denote by $\mathbb X _{\bar P}\subset V$ the subscheme defined by the ideal generated by $ \bar P$. We show the…

Let $X^n\subset C^{n+a}$ or $X^n\subset P^{n+a}$ be a patch of an analytic submanifold of an affine or projective space, let $x\in X$ be a general point, and let L^k be a linear space of dimension k osculating to order m at x. If m is large…

This paper is devoted to understanding curves $X$ over a number field $k$ that possess infinitely many solutions in extensions of $k$ of degree at most $d$; such solutions are the titular low degree points. For $d=2,3$ it is known (by the…

In this paper, we derive a tight upper bound for the size of an intersecting $k$-Sperner family of subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a $q$-analogue of the…

A variety $X$ is covered by lines if there exist a finite number of lines contained in $X$ passing through each general point. I prove two theorems. Theorem 1:Let $X^n\subset P^M$ be a variety covered by lines. Then there are at most $n!$…

A projective hypersurface $X \subseteq \mathbb P^n$ has defect if $h^i(X) \neq h^i(\mathbb P^n)$ for some $i \in \{n, \dots, 2n-2\}$ in a suitable cohomology theory. This occurs for example when $X \subseteq \mathbb P^4$ is not $\mathbb…

We study subvarieties of very general complete intersections $X\subset \mathbb{P}^n$ of multidegree $(d_1,\dots,d_c)$, when $d:= d_1+\dots +d_c$ is sufficiently large. In a seminal paper Ein proved that if $d\geq 2n-c-k+2$, any…

It is well known that the only surfaces that are simultaneously minimal in $\mathbb{R}^3$ and maximal in $\mathbb{L}^3$ are open pieces of helicoids (in the region in which they are spacelike) and of spacelike planes (O. Kobayashi 1983).…

Let $\mathbb{K}$ be an algebraically closed field, and $A \subset \mathbb{K}[x_{1}, \ldots, x_n]$ be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of $\mathbb{K}$-algebras \[…

We prove the following: (a) Let X be a smooth, codimension two subvariety of P6. If X lies on a hyperquintic or if deg(X)<74, then X is a complete intersection. (b) Let X be a smooth, subcanonical threefold in P5. If X lies on a…

In this paper we introduce a notion of dimension and codimension for every element of a distributive bounded lattice $L$. These notions prove to have a good behavior when $L$ is a co-Heyting algebra. In this case the codimension gives rise…

For a semialgebraic set K in R^n, let P_d(K) be the cone of polynomials in R^n of degrees at most d that are nonnegative on K. This paper studies the geometry of its boundary. When K=R^n and d is even, we show that its boundary lies on the…

We prove that there exist hypersurfaces that contain a given closed subscheme $Z$ of the projective space over a finite field and intersect a given smooth scheme $X$ off of $Z$ smoothly, if the intersection $V = Z \cap X$ is smooth.…

We classify all Mathieu subspaces of ${\rm Mat}_n(K)$ of codimension less than $n$, under the assumption that ${\rm char\,} K = 0$ or ${\rm char\,} K \ge n$. More precisely, we show that any proper Mathieu subspace of ${\rm Mat}_n(K)$ of…

Developing a previous idea of Faltings, we characterize the complete intersections of codimension 2 in P^n, n>=3, over an algebraically closed field of any characteristic, among l.c.i. X, as those that are subcanonical and…

We consider the Fano scheme $F_k(X)$ of $k$--dimensional linear subspaces contained in a complete intersection $X \subset \mathbb{P}^n$ of multi--degree $\underline{d} = (d_1, \ldots, d_s)$. Our main result is an extension of a result of…

We find the minimal dimension for a truncated polynomial algebra over an arbitrary field for which there exists a "non-thin" subalgebra. Moreover, we discuss examples of subalgebras, and count them in low dimensions.