Given a field $k$ of characteristic $0$ and a (not necessarily proper) smooth curve $C$ over $k$, i.e. a geometrically connected, geometrically reduced, smooth, separated scheme of finite type over $k$ of dimension $1$.
How do I define its Euler characteristic? If $C$ is proper, one can consider the Euler characteristic of its structure sheaf $$ \chi(C) = \chi_k(\mathcal{O}_C) = \sum_i (-1)^i \dim_k H^i(C, \mathcal{O}_C) = \dim_k H^0(C, \mathcal{O}_C) - \dim_k H^1(C, \mathcal{O}_C) $$ Since in our situation, $H^0(C, \mathcal{O}_C) = k$, setting $g(C) = \dim_k H^1(C, \mathcal{O}_C)$ gives the formula $$ \chi(C) = 1 - g(C) $$ What do I do if $C$ is not proper? I expect some connection with the smooth compactification $\bar C$ of $C$ as in the topological case. To be more precise: If $S$ is a (Riemann) surface obtained from a compact surface of genus $g$ with $n$ points removed, the Euler characteristic of $S$ is $$ \chi(S) = 2 -2 g - n $$ Now I expect a similar formula for $C$, something like $$ \chi(C) = 1 - g(\bar C) - \#(\bar C \setminus C) = \chi(\bar C) - \#(\bar C \setminus C) $$ which suspiciously looks like some excision formula for the Euler characteristic.
If $k = \mathbb{C}$, I would expect that $2 \chi(C) = \chi(C^{\mathrm{an}})$ where $C^{\mathrm{an}}$ denotes the associated Riemann surface. Hence I wonder whether there might be a factor of $\frac{1}{2}$ missing in front of the $\#(\bar C \setminus C)$ (which strikes me as strange, since $\chi(C)$ should be an integer).
To summarize:
- What is a sensible definition of the Euler characteristic $\chi(C)$ of a curve in the non-proper case?
- Is $\chi(C)$ related to the Euler characteristic of the (smooth) compactification?
- If $k = \mathbb{C}$, does $2 \chi(C) = \chi(C^{\mathrm{an}})$ hold?