Related papers: A Generalized Torelli Theorem

In this paper we prove a generalization of Montel's theorem for a class of first order elliptic equations with measurable coefficients involving Hodge-Dirac operators. We then apply this result to sequences of solutions of second order…

Let $\Sigma_{g'}\to \Sigma_g$ be a cover of an orientable surface of genus g by an orientable surface of genus g', branched at n points, with Galois group H. Such a cover induces a virtual action of the mapping class group…

We study the space ${\mathcal X}^{d}(G)$ of pictures of a graph $G$ in complex projective $d$-space. The main result is that the homology groups (with integer coefficients) of ${\mathcal X}^{d}(G)$ are completely determined by the Tutte…

We consider the generalised Jacobian $J_{0}(N)_{\mathbf{m}}$ of the modular curve $X_{0}(N)$ of level $N$, with respect to the modulus $\mathbf{m}$ consisting of all cusps on the modular curve. When $N$ is odd, we determine the group…

In the paper we consider the Jacobian locus $\overline{J_g}$ and the Prym locus $\overline{P_{g+1}}$, in the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$, for $g\geq 7$, and we study the extrinsic geometry…

We obtain new information about divisors on the $d-$th symmetric power $C_{d}$ of a general curve $C$ of genus $g \geq 4.$ This includes a complete description of the effective cone of $C_{g-1}$ and a partial computation of the volume…

We prove that the cohomology of the moduli space of morphisms of a fixed finite degree from a smooth projective curve $C$ of genus $g$ to a complete simplicial toric variety $\mathbb{P}(\Sigma)$, denoted by the rational polyhedral fan…

We approach the Torelli problem of recostructing a curve from its Jacobian from a computational point of view. Following Dubrovin, we design a machinery to solve this problem effectively, which builds on methods in numerical algebraic…

Let G be a discrete group. We give methods to compute for a generalized (co-)homology theory its values on the Borel construction (EG x X)/G of a proper G-CW-complex X satisfying certain finiteness conditions. In particular we give formulas…

We show the geometric syzygy conjecture in positive characteristic. Specifically, if C is a general smooth curve of genus g defined over an algebraically closed field of characteristic p, then all linear syzygy spaces are spanned by…

Let $C$ be a smooth plane curve. A point $P$ in the projective plane is said to be Galois with respect to $C$ if the function field extension induced from the point projection from $P$ is Galois. We denote by $\delta(C)$ (resp.…

We introduce deformation theoretic methods for determining when a curve $X$ in a non-hyperelliptic jacobian $JC$ will deform with $JC$ to a non-jacobian. We apply these methods to a particular class of curves in symmetric powers $C^{(e)}$…

We prove the Lipman-Zariski conjecture for complex surface singularities with $p_g - g - b \le 2$. Here $p_g$ is the geometric genus, $g$ is the sum of the genera of the exceptional curves and $b$ is the first Betti number of the dual…

For any integer $d \geq 1$, we verify the Jacobian Conjecture for a $d$-linear map in two variables. We prove that almost all the coefficients of the formal inverse are in the ideal specified by the Jacobian condition. We find expressions…

Consider a $1$-dimensional centered Gaussian process $W$ with $\alpha$-H\"older continuous paths on the compact intervals of $\mathbb R_+$ ($\alpha\in ]0,1[$) and $W_0 = 0$, and $X$ the local solution in rough paths sense of Jacobi's…

We obtain certain algebraic invariants relevant to study codes on subgroups of weighted projective tori inside an $n$-dimensional weighted projective space. As application, we compute all the main parameters of generalized toric codes on…

In this paper, we prove that for any odd prime $p$ and for any $p$-integer $\alpha $,we have $ \binom{\alpha p-1}{p-1}\equiv 1-\alpha (\alpha -1)(\alpha ^{2}-\alpha -1)p\sum_{k=1}^{p-1}\frac{1}{k}+\alpha ^{2} (\alpha-1)^{2}p^{2}\sum_{1\leq…

Let $G$ be a simply connected semisimple algebraic group over $\mathbb{C}$ and let $\rho :G\rightarrow GL(V_\lambda)$ be an irreducible representation of highest weight $\lambda$. Suppose that $\rho$ has finite kernel. Springer defined…

We study the behavior of the greatest common divisor of a^k-1 and b^k-1, where a,b are fixed integers or polynomials, and k varies. In the integer case, we conjecture that when a and b are multiplicatively independent and in addition a-1…

Let $G$ be a $t$-uniform hypergraph, and let $c(G)$ denote the cyclic index of the adjacency tensor of $G$. Let $m,s,t$ be positive integers such that $t \ge 2$, $s \ge 2$ and $m=st$. The generalized power $G^{m,s}$ of $G$ is obtained from…