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This is question from Gathmann's notes on Algebraic Geometry.

Let $$C_n=\{(x,y)\in\Bbb{C}^2;y^2=(x-1)(x-2)\dots(x-2n)\}\subset\Bbb{C}^2$$

Gathmann says that if we go in a circle around any of the points $1,2,3\dots,n$, "we go from one copy of the plane to another".

I don't know what that means. $y$ clearly has two values for every single value of $x$. So this might mean that we travel from one value of $y$ to another. I can picture this happening for the polynomial $C_1$, but the picture is too complicated for me for $C_r$ where $r>1$.

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Let $p\in C_n$ be a point with $y=0,$ i.e. $p=(k,0)$ for some $k=1,\ldots,2n$. In a small neighborhood $U$ around $p$, all the terms $(x-1),\ldots,(x-2n)$, except for $(x-k)$ don't change dramatically, so inside $U$ $C_n$ actually looks pretty much like $C_1$.

Amitai Yuval
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  • Doesn't $y=0$ imply $y=(0,0)$, and not $(k,0)$? –  Sep 01 '14 at 09:05
  • I just meant that $(k,0)$ would satisfy the equation for instance, where $y=0$. $y\neq (k,0)$. Thanks for your answer. Correcting that minor glitch would be useful for all future readers. –  Sep 01 '14 at 09:15
  • Right, I hadn't noticed my typo. It's corrected now, thanks. Is the answer clear? – Amitai Yuval Sep 01 '14 at 09:18