Related papers: Subbundles of maximal degree

We prove the $p$-curvature conjecture for rank two vector bundles with connection on generic curves, by combining deformation techniques for families of varieties and topological arguments.

Let $X$ be a smooth projective curve of genus $g$, defined over an algebraically closed field $k$, and let $G$ be a connected reductive group over $k$. We say that a $G$-torsor is essentially finite if it admits a reduction to a finite…

In this paper, we prove the degree upper bound of projective subschemes in terms of the reduction number and show that the maximal cases are only arithmetically Cohen-Macaulay subschemes with linear resolution. Furthermore, it can be shown…

Using an Euclidean approach, we prove a new upper bound for the number of closed points of degree 2 on a smooth absolutely irreducible projective algebraic curve defined over the finite field $\mathbb F\_q$.This bound enables us to provide…

We find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our…

Given a vector space $V$ of homogeneous polynomials of the same degree over an infinite field, consider a generic subspace $W$ of $V$. The main result of this paper is a lower-bound (in general sharp) for the dimensions of the spaces…

We give an asymptotic formula for the number of $\mathbb{F}_{q}$-rational points over a fixed determinant moduli space of stable vector bundles of rank $r$ and degree $d$ over a smooth, projective curve $X$ of genus $g \geq 2$ defined over…

We completely describe the components of expected dimension of the Hilbert Scheme of rational curves of fixed degree $k$ in the moduli space ${\rm SU}_{C}(r,L)$ of semistable vector bundles of rank $r$ and determinant $L$ on a curve $C$. We…

For a given genus $g \geq 1$, we give lower bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over ${\mathbb F}_q$. As a consequence of Katz-Sarnak theory, we first get for any…

We show that a generic real projective $n$-dimensional hypersurface of odd degree $d$, such that $4(n-2)=\binom{d+3}3$, contains "many" real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, $d^3\log d$,…

Let SU(r,d) be the moduli space of rank r, degree d vector bundles over a smooth projective curve of genus $g\ge 2$. If (r,d)=1 and d divides r+1, then SU is rational. Furthermore, if $0<\delta < r$ and all prime divisors of $\delta$ divide…

Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\leq e^4-e^3$ whenever $e>1$. This bound is best possible and improves on…

For a line bundle L on a smooth surface S, it is now known that the degree of the Severi variety of cogenus-d curves is given by a universal polynomial in the Chern classes of L and S if L is d-very ample. For S rational, we relax the…

Let $C$ be a smooth projective curve over $\mathbb C$ of genus $g\geqslant 1$. Let $E$ be a vector bundle on $C$ of rank $r$ and degree $e$. Given integers $k_1,k_2,d_1,d_2$ such that $r>k_1>k_2>0$, let $\mathcal Q^{k_1,k_2}_{d_1,d_2}(E)$…

For each natural number $n$ we determine, both asymptotically and exactly, the maximum number of edges an induced subgraph of order $n$ of the $d$-dimension a grid graph ${\ints}^d$ can have. The asymptotic bound is obtained by using a…

Let $(A, \mathfrak m)$ be a normal two-dimensional local ring and $I$ an $\mathfrak m$-primary integrally closed ideal with a minimal reduction $Q$. Then we calculate the numbers: $\mathrm{nr}(I) = \min\{n \;|\; \overline{I^{n+1}} =…

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice…

Let G be a simple graph without isolated vertices. For a vertex i in G, the degree d_i is the number of vertices adjacent to i and the average 2-degree m_i is the mean of the degrees of the vertices which are adjacent to i. The sequence of…

For a projective nonsingular curve of genus $g$, the Brill-Noether locus $W^r_d(C)$ parametrizes line bundles of degree $d$ over $C$ with at least $r+1$ sections. When the curve is generic and the Brill-Noether number $\rho(g,r,d)$ equals…

We consider the moduli space of rank two, odd degree, semi-stable Real vector bundles over a real curve, calculating the singular cohomology ring in odd and zero characteristic for most examples.