We consider the linear system $|2{\Theta }_0|$ of second order theta functions over the Jacobian $JC$ of a non-hyperelliptic curve $C$. A result by J.Fay says that a divisor $D\in |2{\Theta }_0|$ contains the origin $𝒪\in JC$ with multiplicity $4$ if and only if $D$ contains the surface $C-C=\left\{𝒪\right(p-q)\mid p,q\in C\}\subset JC$. In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$, divisors containing the fourfold $C_2-C_2=\{𝒪(p+q-r-s)\mid p,q,r,s\in C\}$, and divisors singular along $C-C$, using...

Consider the families of curves $C^{n,A}:y²=xⁿ+Ax$ and $C_{n,A}:y²=xⁿ+A$ where A is a nonzero rational. Let $J^{n,A}$ and $J_{n,A}$ denote their respective Jacobian varieties. The torsion points of $C^{3,A}\left(\mathbb{Q}\right)$ and $C_{3,A}\left(\mathbb{Q}\right)$ are well known. We show that for any nonzero rational A the torsion subgroup of $J^{7,A}\left(\mathbb{Q}\right)$ is a 2-group, and for A ≠ 4a⁴,-1728,-1259712 this subgroup is equal to $J^{7,A}\left(\mathbb{Q}\right)\left[2\right]$ (for a excluded values of A, with the possible exception of A = -1728, this group has a point of order 4). This is a variant of the corresponding results for $J^{3,A}$ (A ≠ 4) and $J^{5,A}$. We...

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 $X$ be a smooth projective curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p\>0$. 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 $X$.

Let $k$ be a field of characteristic $p\>0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that...

Let $k$ be an algebraically closed field of characteristic $p\>0$. We study obstructions to lifting to characteristic $0$ the faithful continuous action $\phi$ of a finite group $G$ on $k\left[\right[t\left]\right]$. To each such $\phi$ a theorem of Katz and Gabber associates an action of $G$ on a smooth projective curve $Y$ over $k$. We say that the KGB obstruction of $\phi$ vanishes if $G$ acts on a smooth projective curve $X$ in characteristic $0$ in such a way that $X/H$ and $Y/H$ have the same genus for all subgroups $H\subset G$. We determine for which $G$ the KGB...

Let $G$ be a unitary group defined over a non-Archimedean local field of odd residue characteristic and let $H$ be the centralizer of a semisimple rational Lie algebra element of $G.$ We prove that the Bruhat-Tits building ${𝔅}^1\left(H\right)$ of $H$ can be affinely and $G$-equivariantly embedded in the Bruhat-Tits building ${𝔅}^1\left(G\right)$ of $G$ so that the Moy-Prasad filtrations are preserved. The latter property forces uniqueness in the following way. Let $j$ and $j^{\prime }$ be maps from ${𝔅}^1\left(H\right)$ to ${𝔅}^1\left(G\right)$ which preserve the Moy–Prasad filtrations....

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.

We describe all ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝Q{T}^{*}$ transforming projectable-projectable classical torsion-free linear connections $\nabla$ on fibred-fibred manifolds $Y$ into classical linear connections $D\left(\nabla \right)$ on cotangent bundles $T^{*}Y$ of $Y$. We show that this problem can be reduced to finding ${ℱ}^2{ℳ}_{m_1,m_2,n_1,n_2}$-natural operators $D:Q_{proj-proj}^{\tau }⇝(T^{*},{\otimes }^p{T}^{*}\otimes {\otimes }^{q}T)$ for $p=2$, $q=1$ and $p=3$, $q=0$.

Let $f\left(x\right)$ be a cubic, monic and separable polynomial over a field of characteristic $p\ge 3$ and let $E$ be the elliptic curve given by $y^2=f\left(x\right)$. In this paper we prove that the coefficient at $x^{\frac{1}{2}p(p-1)}$ in the $p$–th division polynomial of $E$ equals the coefficient at $x^{p-1}$ in $f{\left(x\right)}^{\frac{1}{2}(p-1)}$. For elliptic curves over a finite field of characteristic $p$, the first coefficient is zero if and only if $E$ is supersingular, which by a classical criterion of Deuring (1941) is also equivalent to the vanishing of the second coefficient. So the...

A monomial self-map $f$ on a complex toric variety is said to be $k$-stable if the action induced on the $2k$-cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of $f$, we can find a toric model with at worst quotient singularities where $f$ is $k$-stable. If $f$ is replaced by an iterate one can find a $k$-stable model as soon as the dynamical degrees ${\lambda }_k$ of $f$ satisfy ${\lambda }_k^2\>{\lambda }_{k-1}{\lambda }_{k+1}$. On the other hand, we give examples of monomial maps $f$, where...

Let $N$ be a normal subgroup of a group $G$. The structure of $N$ is given when the $G$-conjugacy class sizes of $N$ is a set of a special kind. In fact, we give the structure of a normal subgroup $N$ under the assumption that the set of $G$-conjugacy class sizes of $N$ is $(p_{1n_1}^{a_{1n_1}},\cdots ,p_{11}^{a_{11}},1)\times \cdots \times (p_{r{n}_r}^{a_{r{n}_r}},\cdots ,p_{r1}^{a_{r1}},1)$, where $r>1$, $n_i>1$ and $p_{ij}$ are distinct primes for $i\in \{1,2,\cdots ,r\}$, $j\in \{1,2,\cdots ,n_i\}$.