- Show that $\PP^n\times \PP^m$ is rational
  
> [!solution]- Solution
> 
> $\left[x_0: x_1: \ldots: x_n\right] \times\left[y_0: y_1: \ldots: y_m\right] \rightarrow\left[1: \frac{x_1}{x_0}: \frac{x_2}{x_0}: \ldots: \frac{x_n}{x_0}: \frac{y_1}{y_0}: \frac{y_2}{y_0}, \ldots: \frac{y_n}{y_0}\right]$
> This has inverse
> $\left[1: z_1: \ldots: z_{n+m}\right] \rightarrow\left[1: z_1: \ldots: z_n\right] \times\left[1: z_{n+1}: \ldots: z_{n+m}\right]$
> 


- What is the maximum number of ramification points that a mapping of finite degree from one smooth projective curve over $\mathbb{C}$ of genus 1 to another (smooth projective curve of genus 1 ) can have? Give an explanation for your answer.
  
> [!solution]- Solution
> By the Riemann-Hurwitz formula, if we have a mapping $f$ of finite degree $d$ from one smooth projective (irreducible, say) curve onto another the Euler characteristic of the source curve is $d$ times the Euler characteristic of the target minus a certain nonnegative number $e$, and moreover $e$ is zero if and only if the mapping is unramified. Now compute: the Euler characterstic of our source and target curves is, by hypothesis, 0 and so this $e$ is zero, and therefore the mapping is unramified.


- Is every smooth projective curve of genus 0 defined over the field of complex numbers isomorphic to a conic in the projective plane? Give an explanation for your answer.

> [!solution]- Solution
Solution (Sketch). Yes. Apply the Riemann-Roch theorem which guarantees the existence of a nonconstant meromorphic function with a simple pole at exactly one point. Argue that this meromorphic function identifies the curve with $\mathbb{P}^1$, and using that fact, embed the curve as a conic in the plane in any convenient way, e.g., If $t_0, t_1$ are projective $\left(\mathbb{P}^1\right)$ coordinates, let $z_0=t_0^2$, $z_1=t_0 t_1 z_2=t_1^2$ be the map to $\mathbb{P}^2$. The conic, then, would be $z_0 z_2=z_1^2$. (Alternatively: one can consider the complete linear system attached to the anticanonical divisor.)