In this question, all varieties are over an algebraically closed field $k$ of characteristic zero.

What does it mean for a projective variety to be smooth? Give an example of a smooth affine variety $X \subset \mathbb{A}_{k}^{n}$ whose projective closure $\bar{X} \subset \mathbb{P}_{k}^{n}$ is not smooth.

What is the genus of a smooth projective curve? Let $X \subset \mathbb{P}_{k}^{4}$ be the hypersurface $V\left(X_{0}^{3}+X_{1}^{3}+X_{2}^{3}+X_{3}^{3}+X_{4}^{3}\right)$. Prove that $X$ contains a smooth curve of genus $1 .$

Let $C \subset \mathbb{P}_{k}^{2}$ be an irreducible curve of degree 2 . Prove that $C$ is isomorphic to $\mathbb{P}_{k}^{1}$.

We define a generalized conic in $\mathbb{P}_{k}^{2}$ to be the vanishing locus of a non-zero homogeneous quadratic polynomial in 3 variables. Show that there is a bijection between the set of generalized conics in $\mathbb{P}_{k}^{2}$ and the projective space $\mathbb{P}_{k}^{5}$, which maps the conic $V(f)$ to the point whose coordinates are the coefficients of $f$.

(i) Let $R^{\circ} \subset \mathbb{P}_{k}^{5}$ be the subset of conics that consist of unions of two distinct lines. Prove that $R^{\circ}$ is not Zariski closed, and calculate its dimension.

(ii) Let $I$ be the homogeneous ideal of polynomials vanishing on $R^{\circ}$. Determine generators for the ideal $I$.