Related papers: Linear subspaces of minimal codimension in hypersu…

We prove that for any cubic polynomial of slice rank $r$, the intersection of all linear subspaces of minimal codimension contained in the corresponding hypersurface has codimension $\le r^2+\frac{(r+1)^2}{4}+r$ in the affine space. This is…

Fix positive integers $n,r,d$. We show that if $n,r,d$ satisfy a suitable inequality, then any smooth hypersurface $X\subset \mathbb{P}^n$ defined over a finite field of characteristic $p$ sufficiently large contains a rational $r$-plane.…

Let $k$ be any field. Let $X \subset \mathbb{P}_k^N$ be a degree $d \geq 2$ hypersurface. Under some conditions, we prove that if $X(K) \neq \emptyset$ for some extension $K/k$ with $n:=[K:k] \geq 2$ and $\gcd(n,d)=1$, then $X(L) \neq…

Ballico proved that a smooth projective variety $X$ of degree $d$ over a finite field of $q$ elements admits a smooth hyperplane section if $q\geq d(d-1)^{\dim X}$. In this paper, we refine this criterion for higher codimensional linear…

We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the…

Let K be an arbitrary (commutative) field with at least three elements, and let n, p and r be positive integers with r<=min(n,p). In a recent work, we have proved that an affine subspace of M_{n,p}(K) containing only matrices of rank…

We classify hypersurfaces of the Minkowski space $\L^{n+1}$ that carry a totally geodesic foliation with complete leaves of codimension one. We prove that such a hypersurface is ruled, or a partial tube over a curve or contains a two or…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

We ask when certain complete intersections of codimension $r$ can lie on a generic hypersurface in $\PP^n$. We give a complete answer to this question when $2r \leq n+2$ in terms of the degrees of the hypersurfaces and of the degrees of the…

Let $K$ be a finitely generated field. We construct an $n$-dimensional linear system $\mathcal{L}$ of hypersurfaces of degree $d$ in $\mathbb{P}^n$ defined over $K$ such that each member of $\mathcal{L}$ defined over $K$ is smooth, under…

Let $d$ and $n$ be positive integers, and $E/F$ be a separable field extension of degree $m=\binom{n+d}{n}$. We show that if $|F| > 2$, then there exists a point $P\in \mathbb{P}^n(E)$ which does not lie on any degree $d$ hypersurface…

K be a field and let m and n be positive integers, where m does not exceed n. We say that a non-zero subspace of m x n matrices over K is a constant rank r subspace if each non-zero element of the subspace has rank r, where r is a positive…

Let $n,p,r$ be positive integers with $n \geq p\geq r$. A rank-$\overline{r}$ subset of $n$ by $p$ matrices (with entries in a field) is a subset in which every matrix has rank less than or equal to $r$. A classical theorem of Flanders…

Asgarli, Ghioca, and Reichstein proved that if $K$ is a field with $|K|>2$, then for any positive integers $d$ and $n$, and separable field extension $L/K$ with degree $m=\binom{n+d}{d}$, there exists a point $P\in \mathbb{P}^n(L)$ which…

We prove that a component of the closure of the set of star points on a hypersurface X of degree d>2 in N-dimensional projective space is linear. Afterwards, we focus on the case where the component is of maximal dimension N-2 and the case…

Let $X\subset \mathbb{P}^{2k+1}$ be a smooth hypersurface containing two k-dimensional linear spaces $\Pi_1,\Pi_2$ intersecting in codimension one. In this paper we study the question whether the Hodge loci $NL([\Pi_1]+\lambda[\Pi_2])$ and…

Consider the Fano scheme $F_k(Y)$ parameterizing $k$-dimensional linear subspaces contained in a complete intersection $Y \subset \mathbb{P}^m$ of multi-degree $\underline{d} = (d_1, \ldots, d_s)$. It is known that, if $t := \sum_{i=1}^s…

We prove that, under mild restrictions, the space of codimension-one foliations of degree one on a smooth projective complete intersection has two irreducible components of logarithmic type. We also prove that the same conclusion holds for…

Let $X$ be an arbitrary smooth hypersurface in $\mathbb{C} \mathbb{P}^n$ of degree $d$. We prove the de Jong-Debarre Conjecture for $n \geq 2d-4$: the space of lines in $X$ has dimension $2n-d-3$. We also prove an analogous result for…

R. Beheshti showed that, for a smooth Fano hypersurface $X$ of degree $\leq 8$ over the complex number field $\mathbb{C}$, the dimension of the space of lines lying in $X$ is equal to the expected dimension. We study the space of conics on…