Avatar The Game Keygen Generator Free 12


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Convergence of $k$-fold Jacobians of cyclic covers of the projective line

Let $X$ be a smooth curve and $k \in \mathbb{N}$. Let $f: Y \to X$ be a $k$-cyclic cover branched over $k$ distinct points on $X$. The sequence of Jacobians of $f$:
J_Y^k := f_\ast( \Omega_Y^k)
is a coherent sheaf on $X$ which is the $k$-fold product of the Jacobian of the smooth curve $f(Y)$. Furthermore, if the generic fiber is smooth, then $J_Y^k$ can be calculated as the tensor product of $k$ copies of the jacobian of $f(Y)$.
What can be said about the asymptotic behaviour of the sequence of “Jacobians of $Y$” in terms of sections of $J_Y^k$? For example, does the sequence of rank of $J_Y^k$ sublattices $\mathrm{rk}(\Lambda)$, where $\Lambda$ runs through all sublattices of $J_Y^k$, grow when the branch points are taken in “nice” closed subsets of $X$ (e.g. finite sets or 0-dimensional subsets), when $X$ is a complex elliptic curve, and so on?


It is clear from your description that your sequence $\text{rk}\,\Lambda$ of subspaces of $\text{Pic}^0\,Y$ is just a finite product of extensions of $H^0(X,\mathcal O_X)$ by $\mathbf

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