Exercise 110.65.6 (Fixed points). Let $k$ be an algebraically closed field.

1. If $G = \mathbf{G}_{m, k}$ show that if $G$ acts on a projective variety $X$ over $k$, then the action has a fixed point, i.e., prove there exists a point $x \in X(k)$ such that $a(g, x) = x$ for all $g \in G(k)$.

2. Same with $G = (\mathbf{G}_{m, k})^ n$ equal to the product of $n \geq 1$ copies of the multiplicative group.

3. Give an example of an action of a connected group scheme $G$ on a smooth projective variety $X$ which does not have a fixed point.

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