Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a number field $K$ with good reduction outside a finite set of places $S$ of $K$. We prove that $C$ has a Weierstrass model over the ring of integers of $K$ with height effectively bounded only in terms of $g$, $S$ and $K$. In particular, we obtain that for any given number field $K$, finite set of places $S$ of $K$ and integer $g\ge 1$ one can in principle determine the set of $K$-isomorphism classes of hyperelliptic curves over $K$ of genus $g$ with good reduction outside...

Let $C$ be a smooth curve of genus $g$. For each positive integer $r$ the birational $r$-gonality $s_r\left(C\right)$ of $C$ is the minimal integer $t$ such that there is $L\in {\text{Pic}}^t\left(C\right)$ with $h^0(C,L)=r+1$. Fix an integer $r\ge 3$. In this paper we prove the existence of an integer $g_r$ such that for every integer $g\ge g_r$ there is a smooth curve $C$ of genus $g$ with $s_{r+1}\left(C\right)/(r+1)>s_r\left(C\right)/r$, i.e. in the sequence of all birational gonalities of $C$ at least one of the slope inequalities fails.

In this paper, we examine a natural question concerning the divisors of the polynomial $x^n-1$: “How often does $x^n-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when $x^n-1$ is factored in $\mathbb{Z}\left[x\right]$. In this paper, we replace $\mathbb{Z}\left[x\right]$ with ${𝔽}_p\left[x\right]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.

The notion of $\beta$-normality was introduced and studied by Arhangel’skii, Ludwig in 2001. Recently, almost $\beta$-normal spaces, which is a simultaneous generalization of $\beta$-normal and almost normal spaces, were introduced by Das, Bhat and Tartir. We introduce a new generalization of normality, namely weak $\beta$-normality, in terms of $\theta$-closed sets, which turns out to be a simultaneous generalization of $\beta$-normality and $\theta$-normality. A space $X$ is said to be weakly $\beta$-normal (w$\beta$-normal$)$ if for every...

Let $G$ be a finite group and $p$ a prime number. We prove that if $G$ is a finite group of order $\left|\mathrm{PSL}\right(2,p^2\left)\right|$ such that $G$ has an irreducible character of degree $p^2$ and we know that $G$ has no irreducible character $\theta$ such that $2p\mid \theta \left(1\right)$, then $G$ is isomorphic to $\mathrm{PSL}(2,p^2)$. As a consequence of our result we prove that $\mathrm{PSL}(2,p^2)$ is uniquely determined by the structure of its complex group algebra.

Let $(M,\theta )$ be a compact CR manifold of dimension $2n+1$ with a contact form $\theta$, and $L=(2+2/n){\Delta }_b+R$ its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form $\tilde{\theta }$ on $M$ conformal to $\theta$ which has a constant Webster curvature. This problem is equivalent to the existence of a function $u$ such that $Lu=u^{1+2/n}$, $u>0$ on $M$. D. Jerison and J. M. Lee solved the CR Yamabe problem in the case where $n\ge 2$ and $(M,\theta )$ is not locally CR equivalent to the sphere $S^{2n+1}$ of ${𝐂}^n$. In a join work with R. Yacoub, the CR Yamabe...

Let $D$ be a ${𝒞}^d$ $q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More precisely, we prove that for a $\overline{\partial }$-closed form $f$ in ${𝒞}_{0,q}^k(X\setminus D,E)$, $1\le q\le n-2$, $n\ge 3$, with compact support and for $\epsilon$ with $0<\epsilon <1$ there...

Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image ${𝒩}_1(D,X)$ of ${𝒩}_1\left(D\right)$ in ${𝒩}_1\left(X\right)$ under the natural push-forward of $1$-cycles. We show that ${\rho }_X-{\rho }_D\le codim{𝒩}_1(D,X)\le 8$. Moreover if $codim{𝒩}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $codim{𝒩}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that ${\rho }_X-{\rho }_T=4$.

In this paper we prove the following theorems in incidence geometry. 1. There is $\delta >0$ such that for any $P_1,\cdots ,P_4$, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le n^{(1+\delta )/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then $P_1,\cdots ,P_4$ are collinear. If the number of the distinct lines is $<c{n}^{1/2}$ then the cross ratio of the four points is algebraic. 2. Given $c>0$, there is $\delta >0$ such that for any $P_1,P_2,P_3\in {ℂ}^2$ noncollinear, and $Q_1,\cdots ,Q_n\in {ℂ}^2$, if there are $\le c{n}^{1/2}$ many distinct lines between $P_i$ and $Q_j$ for all $i$, $j$, then for any $P\in {ℂ}^2\setminus \{P_1,P_2,P_3\}$, we have $\delta n$ distinct lines between $P$ and $Q_j$. 3. Given...

Let $X$ be a smooth projective curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p\&gt;0$. Given a semistable vector bundle $E$ over $X$, we show that its direct image $F_{*}E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank-$p$ vector bundles $F_{*}L$, where $L$ is a line bundle over $X$.