Related papers: The resolution of the universal Abel map via tropi…

We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem…

Over the moduli space of smooth curves, the double ramification cycle can be defined by pulling back the unit section of the universal jacobian along the Abel-Jacobi map. This breaks down over the boundary since the Abel-Jacobi map fails to…

For each universal genus-$g$ polarization $\mu$ of degree $d$, we construct a universal tropical Jacobian $J_{\mu,g}^{trop}$ as a generalized cone complex over the moduli space of stable pointed genus-$g$ tropical curves. We show several…

In the classical case of irreducible smooth algebraic curves every genus $2$ curve is hyperelliptic, or in other words there is a complete linear series $g_2^1$ on them. On the other hand if $g > 2$, then a generic smooth curve of genus $2$…

Let $C$ be a nodal curve and $L$ be an invertible sheaf on $C$. Let $\alpha_{L}:C\dashrightarrow J_{C}$ be the degree-$1$ rational Abel map, which takes a smooth point $Q\in C$ to $\left[ m_{Q}\otimes L\right] $ in the Jacobian of $C$. In…

Let $A=(a_1,\ldots,a_n)$ be a vector of integers with $d=\sum_{i=1}^n a_i$. By partial resolution of the classical Abel-Jacobi map, we construct a universal twisted double ramification cycle $\mathsf{DR}^{\mathsf{op}}_{g,A}$ as an…

We generalize Abel's classical theorem on linear equivalence of divisors on a Riemann surface. For every closed submanifold $M^d \subset X^n$ in a compact oriented Riemannian $n$--manifold, or more generally for any $d$--cycle $Z$ relative…

Recently, the first Abel map for a stable curve of genus g>1 has been constructed. Fix an integer d>0 and let C be a stable curve of compact type of genus g>1. We construct two d-th Abel maps for C, having different targets, and we compare…

Let $A=(a_1,\ldots, a_n)$ be a vector of integers which sum to $k(2g-2+n)$. The double ramification cycle $\mathsf{DR}_{g,A}\in \mathsf{CH}^g(\mathcal{M}_{g,n})$ on the moduli space of curves is the virtual class of an Abel-Jacobi locus of…

In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian…

We solve the problem of inversion of an extended Abel-Jacobi map $$ \int_{P_{0}}^{P_{1}}\omega +...+\int_{P_{0}}^{P_{g+n-1}}\omega ={\bf z}, \qquad \int_{P_{0}}^{P_{1}}\Omega_{j1}+... +\int_{P_{0}}^{P_{g+n-1}}\Omega_{j1} =Z_{j},\quad…

We construct the Abel-Jacobi map for Mumford curves over any complete non-archimedean field, using multiplicative integrals and in the setting of Berkovich analytic geometry. Along the way, we proof some results concerning graphs and…

In this paper we resolve the degree-2 Abel map for nodal curves. Our results are based on a previous work of the authors reducing the problem of the resolution of the Abel map to a combinatorial problem via tropical geometry. As an…

We construct Abel maps for a stable curve $X$. Namely, for each one-parameter deformation of $X$ with regular total space, and every integer $d>0$, we construct by specialization a map $\alpha^d_X$ from the smooth locus of $X^d$ to the…

We study the center problem for the class $\mathcal E_\Gamma$ of Abel differential equations $\frac{dv}{dt}=a_1 v^2+a_2 v^3$, $a_1,a_2\in L^\infty ([0,T])$, such that images of Lipschitz paths $\tilde A:=\bigl(\int_0^\cdot a_1(s)ds,…

Let X be an Abelian surface and C a holomorphic curve in X representing a primitive homology class. The space of genus g curves in the class of C is g dimensional. We count the number of such curves that pass through g generic points and we…

We develop a general framework for Abel maps associated with a family $X/S$ of integral curves using derived algebraic geometry. For compactified Picard schemes, our approach yields relative quasi-smooth derived enhancements of the Quot…

We define a generalized Jacobian $\mathrm{J}_\mathfrak{m}(\mathit{Gr})$ and a generalized Picard group $\mathrm{P}_\mathfrak{m}(\mathit{Gr})$ of a graph $\mathit{Gr}$ with respect to a modulus $ \mathfrak{m}=\sum_{i=1}^s m_iw_i$ with $w_i$…

In this paper we investigate Abel maps on normal surface singularities described in \cite{NNI}. We investigate the affine version of the class of the images of Abel maps on normal surface singularities. More precisely we consider the…

Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry,…