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\begin{center}\begin{large}\textbf{Math 137 - Problem Set 6\\
Due Wednesday, Mar 11}\end{large}\end{center}



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All rings are commutative, and $k$ is an algebraically closed field.

\begin{enumerate} 

\item Let $k = \C$. Find the multiple points, and the tangent lines at the multiple points, for each of the following curves:
\begin{enumerate}
\item $y^3-y^2+x^3-x^2+3xy^2+3x^2y +2xy $
\item $x^4+y^4-x^2y^2$
\item $x^3+y^3-3x^2-3y^2+3xy +1$
\end{enumerate}

\item The \textbf{\textit{ring of formal power series over}} $k$ is defined $$k[[x]] := \left\{ \sum_{i=0}^\infty a_i x^i \mid a_i \in k \right\}.$$ Show that $k[[x]]$ is a DVR with uniformizing parameter $x$.


\item If a curve $f$ of degree $n$ has a point $P$ of multiplicity $n$, show that $f$ consists of
$n$ lines through $P$ (not necessarily distinct).

\item Let $T: \A^2 \to \A^2$ be a regular map, and $T(Q) = P$. 
\begin{enumerate}
\item If $f \in k[x,y]$, show that the following inequality of multiplicities holds:
$$m_Q(T^*(f)) \geq m_P(f).$$
\item \textbf{(Extra credit)}\ Let $T =(T_1,T_2)$, and define $$J_QT =\begin{pmatrix} \frac{\partial T_1}{\partial x}(Q) &\frac{\partial T_1}{\partial y}(Q)  \\ \frac{\partial T_2}{\partial x}(Q)  &\frac{\partial T_2}{\partial y}(Q)   \end{pmatrix}$$ to be the \textit{\textbf{Jacobian matrix}} of $T$ at $Q$. Show that if $J_QT$ is invertible, $m_Q(T^*(f))= m_P(f)$. Does the converse hold?
\end{enumerate}

\end{enumerate}

\noindent You might want to wait until after class on Monday to work on the next two problems.

\begin{enumerate}
 \setcounter{enumi}{4}

\item Let $P = (0,0)$ and $k = \C$. Consider the following affine plane curves containing $P$:
\begin{itemize}
\item $A = x^2-y$
\item $B = y^2 -x^3+x$
\item $C = y^2 - x^3$
\item $D = y^2 - x^3 - x^2$
\item $E = (x^2+y^2)^3-4x^2y^2$
\end{itemize}

\begin{enumerate}
\item Compute $I_P(A, C)$.
\item Compute $I_P(C , D)$.
\item Compute $I_P(B , E)$.
\end{enumerate}

\item Let $f, g,$ and $h$ be affine plane curves. 

\begin{enumerate}
\item If $P$ is a simple point on $f$, show $$I_P(f ,g+h) \geq \min \{I_P(f, g),I_P(f , h)\}.$$ 

\item Give an example to show that (a) may be false if $P$ is not simple on $f$.
\end{enumerate}

\end{enumerate}

 \noindent \textbf{Bonus questions} (Extra credit)

\begin{enumerate}
 \setcounter{enumi}{6}

\item Let $P = (0,0)$ lie on an irreducible curve $f$. Let $\mathfrak{m}=\mathfrak{m}_P(f)$ be the maximal ideal of $\mathcal{O}_P(f)$. 

\begin{enumerate}

\item Show that $\dim_k (\mathfrak{m}^n/\mathfrak{m}^{n+1})= n +1$ for $0 \leq n < m_P (f ).$

\item Conclude that $P$ is a simple point if and only if $\dim_k (\mathfrak{m}/\mathfrak{m}^2) = 1$; otherwise $\dim_k (\mathfrak{m}/\mathfrak{m}^2) = 2$.
\end{enumerate}

\item Let $X$ and $Y$ be affine varieties, and $\phi:X\to Y$ a regular map. We showed  on the last homework that $X$ and $Y$ are locally ringed spaces, with structure sheaves $\mathcal{O}_X$ and $\mathcal{O}_Y$, respectively. Look up the definition of a morphism between locally ringed spaces (e.g. \href{https://en.wikipedia.org/wiki/Ringed_space#Morphisms}{here}), and show that $\phi$ is a morphism of locally ringed spaces.

\end{enumerate}

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