# Question 1 (7 points):

Let $k$ be an algebraic closed field, and let $X\subset \mathbb{A}^n_k$ and $Y\subset \mathbb{A}^m_k$ be algebraic sets. Let $\varphi:X\to Y$ be a polynomial map and let $\varphi^*:k[Y]\to k[X]$ be the corresponding map between coordinate rings.

1. Prove that if $\varphi$ is surjective, then $\varphi^*$ is injective.
2. Prove that if $\varphi^*$ is surjective, then $\varphi$ is injective.
3. Show that the converses of parts 1 and 2 don’t hold.
4. If $V\subset Y$ is an algebraic subset, prove that $\varphi^{-1}(V)\subset X$ is also an algebraic subset.
5. If $W\subset X$ is an algebraic subset, give an example to show that $\varphi(W)\subset Y$ need not be an algebraic subset.

Hints: Two polynomials are the same if and only if they take on the same value at every point. Two points $p$ and $q$ are different if and only if there is a polynomial that takes on different values at $p$ and $q$. For 3 and 5, consider the map $V(xy-1)\to \mathbb{A}_k^1$ given by $(a,b)\mapsto a$.

# Question 2 (3 points):

The nodal cubic is the algebraic set $X=V(y^2-x^3-x^2)$.

1. Show that the map $\varphi:\mathbb{A}_{\mathbb{C}}^1\to \mathbb{A}^2_{\mathbb{C}}$ given by $\varphi(t)=(t^2-1, t^3-t)$ lands inside $X$.
2. Show that the map $\varphi$ is surjective onto $X$. Where does $\varphi$ fail to be injective?
3. Prove that $X$ is irreducible. Using $\varphi$ or $\varphi^*$ might help…

## Note (not necessary to do the problem, but fun and potentially helpful)

Sketch the real part of $X$ (maybe by considering how it’s related to the graph of $y=x^3+x^2$).

The map $\varphi$ has the following geometric interpretation, which may be enlightening. Consider the line through the origin with slope $t$, i.e., $y=tx$. If we restrict $y^2-x^3-x^2$ to this line, it will be a cubic polynomial in one variable, and hence have three roots. The origin will always be a double root, and hence there will be one more root. Geometrically, this means the line $y=tx$ intersects $X$ at the origin with multiplicity two, and at one other point. This other point is $\varphi(t)$.