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# Bonus Questions

## Irreducibility

1. Let $X \neq \emptyset$ be a topological space. Prove that $X$ is irreducible if and only if all non-empty open subsets of $X$ are connected.
2. Prove that the cuspidal cubic $Y \subset \mathbb{A}_{\mathbb{C}}^2$ of equation $x^3-y^2=0$ is irreducible. (Hint: express $Y$ as image of $\mathbb{A}^1$ in a continuous map...)
3. Give an example of two irreducible subvarieties of $\mathbb{P}^3$ whose intersection is reducible.
4. Find the irreducible components of the following algebraic sets over the complex field:
	a) $V\left(y^4-x^2, y^4-x^2 y^2+x y^2-x^3\right) \subset \mathbb{A}^2$;
	b) $V\left(y^2-x z, z^2-y^3\right) \subset \mathbb{A}^3$.
5. Let $Z$ be a topological space and let $\left\{U_\alpha\right\}_{\alpha \in I}$ be an open covering of $Z$ such that $U_\alpha \cap U_\beta \neq \emptyset$ for $\alpha \neq \beta$ and that all $U_\alpha$ 's are irreducible. Prove that $Z$ is irreducible.

## Calculations


- Show that $\Pic(\CP^1) \cong \ZZ$.
- Compute $H^0(X; \mcf)$ for
	- $X = \PP^1, \mcf = \OO_X$
	- $X = \PP^1, \mcf = \OO_X(m)$
	- $X = \PP^n, \mcf = \OO_X(m)$
	- $X = \PP^n, \mcf = \Omega_{X/k}$
	- All of these with $X = \AA^n$.
	- $X$ a toric variety, $\mcf = \OO(D)$ for $D$ a $T\dash$invariant divisor.
- Describe morphisms $\PP^n \to \AA^m$.
- Describe consequences of the following properties, and give an example and a counterexample of each:
	- Separated varieties/morphisms
	- Proper varieties/morphisms
	- Complete varieties
	- Flat morphisms
	- Reduced schemes
	- Normal varieties
- Show that $\Bl_0 \AA^{n+1} = \OO_{\PP^n}(-1)$.
- Show that $\omega_{\PP^1} = \OO_{\PP^1}(-2)$.
- Define the tautological bundle on $\PP^n$.
	- What is the degree of the canonical line bundle on $\CP^n$? Answer: $-2$.
- Show that $\Cart\Div(X) \cong H^1(X; \OO_X\units)$.
- Produce a Weil divisors that is not Cartier.
- How are line bundles related to Cartier divisors?