Related papers: Complete Linear Series on a Hyperelliptic Curve

We compute the $\integ/\ell$ and $\integ_\ell$ monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the $\integ/\ell$ monodromy of the moduli space of…

In this paper we consider the question of whether there exists a hyperelliptic curve of genus $g$ which is defined over $\FF_q$ but has no rational points over $\FF_q$ for various pairs $(g,q)$.

Consider $E$ a vector bundle over a smooth curve $C$. We compute the $\delta$-invariant of all ample ($\mathbb{Q}$-) line bundles on $\mathbb{P}(E)$ when $E$ is strictly Mumford semistable. We also investigate the case when one assumes that…

Let C be a smooth projective curve with genus g>1 and Clifford index c(C) and let L be a line bundle on C generated by its global sections. The morphism i:C -->P(H^0(L))=P is well-defined and i*T is the restriction to C of the tangent…

Let $C\subset \mathbb {P}^n$ be a smooth curve and $N_C$ its normal bundle. $N_C$ satisfies strong interpolation if for all integers $s>0$ and $\lambda _i\in \{0,1,\dots ,n-1\}$, $1\le i \le s$, there are distinct points $P_1,\dots ,P_s\in…

Let $k$ be an algebraically closed field of characteristic zero, and let $X/k$ be a projective variety. The conjectures of Demailly--Green--Griffiths--Lang posit that every integral subvariety of $X$ is of general type if and only if $X$ is…

Let $k$ be a number field, and let $S$ be a finite set of maximal ideals of the ring of integers of $k$. In his 1962 ICM address, Shafarevich asked if there are only finitely many $k$-isomorphism classes of algebraic curves of a fixed genus…

Consider a fiber bundle in which the total space, the base space and the fiber are all symplectic manifolds. We study the relations between the quantization of these spaces. In particular, we discuss the geometric quantization of a vector…

In this paper, we prove that the admissible canonical bundle of the universal family of curves is a big adelic line bundle, and apply it to prove a uniform Bogomolov-type theorem for curves over global fields of all characteristics. This…

Let $H$ be a Krull monoid with finite class group $G$ and suppose that every class contains a prime divisor. If an element $a \in H$ has a factorization $a=u_1 \cdot \ldots \cdot u_k$ into irreducible elements $u_1, \ldots, u_k \in H$, then…

Let $G$ be a group. The subsets $A_1,\ldots,A_k$ of $G$ form a complete factorization of group $G$ if if they are pairwise disjoint and each element $g\in G$ is uniquely represented as $g=a_1\ldots a_k$, with $a_i\in A_i$. We prove the…

We compare the behaviour of entire curves and integral sets, in particular in relation to locally trivial fiber bundles, algebraic groups and finite ramified covers over semi-abelian varieties.

Let $C$ be a curve in $\mathbb{P}^4$ and $X$ be a hypersurface containing it. We show how it is possible to construct a matrix factorization on $X$ from the pair $(C,X)$ and, conversely, how a matrix factorization on $X$ leads to curves…

Let $(X,L)$ be a polarized complex abelian variety of dimension $g$ where $L$ is a polarization of type $(1,...,1,d)$. For $(X,L)$ genberic we prove the following: (1) If $d \ge g+2$, then $\phi_L\colon X \to {\bf P}^{d-1}$ defines a…

Let $r \geq 2, d$ be two integers which are coprime to each other. Let $C$ be a smooth projective curve of genus $g \geq 2$ and $M(r,L)$ be the moduli space of rank $r$ stable vector bundles on $C$ whose determinants are isomorphic to a…

In this paper we show how to explicitly write down equations of hyperelliptic curves over Q such that for all odd primes l the image of the mod l Galois representation is the general symplectic group. The proof relies on understanding the…

Let $X=G/P$ be a partial flag variety, where $G$ is a semi-simple, simply connected algebraic group defined over an algebraically closed field $K$ of positive characteristic. Let $\mathsf{F}\colon X\to X$ be the absolute Frobenius morphism.…

In this paper we study a family of curves obtained by fibre products of hyperelliptic curves. We then exploit this family to construct examples of curves of given genus g over a finite field Fq with many rational points. The results…

Let X be a smooth genus g curve equipped with a simple morphism f: X -> C, where C is either the projective line or more generally any smooth curve whose gonality is computed by finitely many pencils. Here we apply a method developed by…

Given a symplectic three-fold $(M,\omega)$ we show that for a generic almost complex structure $J$ which is compatible with $\omega$, there are finitely many $J$-holomorphic curves in $M$ of any genus $g\geq 0$ representing a homology class…