I have some familiarity with the notions of modern algebraic geometry, but have little knowledge in the concrete theory of curves, so I am stuck at the following questions. Let $Y$ be a smooth geometrically integral curve over a field $K$. I have seen the Euler characteristic $\chi(Y)$ be defined as follows: if $Y_{\overline{K}}$ is expressed as a smooth projective curve of genus $g$ minus $r$ points, then $\chi(Y):=2-2g-r$. This raises the following questions:
$(1)$ Why can any smooth integral curve over an algebraically closed field be written as a smooth projective curve minus finitely many points? $(2)$ How unique is this description, e.g. is the pair of numbers $(g,r)$ unique or is only the sum $2g+r$ unique?
Now here are my ideas on the matter, though it may be that they lead nowhere. Corollary on page $5$ of http://math.stanford.edu/~vakil/725/class25.pdf says that any nonsingular curve is either projective or affine. (Incidentally, I don't see how this corollary follows from the Proposition stated before it in Vakil's class notes, so an explanation for that would also be appreciated :)) Hence $Y_{\overline{K}}$ is either projective (in which case we are done) or affine, so we may assume that $Y_{\overline{K}}$ is affine, say $Y_{\overline{K}} \subset \mathbb{A}^n$. Then we may take the projective closure of $Y_{\overline{K}}$, i.e. embed $\mathbb{A}^n$ into $\mathbb{P}^n$ (e.g. as the complement of $\{[x_1,...,x_n,0]\}$) and then take the closure $V$ of $Y_{\overline{K}}$ in $\mathbb{P}^n$, which is a projective curve. But I am not sure if $V$ is necessarily smooth, in general smoothness is not preserved by taking the projective closure. Moreover, I'm not sure if the complement $V - Y_{\overline{K}}$ is finitely many points and have no idea as to question $(2)$.
