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\begin{center}\begin{large}\textbf{Math 137 - Problem Set 10\\
Due Friday, Apr 17}\end{large}\end{center}



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

\begin{enumerate} 

\item 
\begin{enumerate}
\item Let $P_1, \ldots, P_r \in \P^2$ and $d \geq 1$. Show that there is a plane curve of degree $d$ that doesn't contain any of the $P_i$.

\item Does (a) generalize to hypersurfaces of degree $d$ in $\P^n$?
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\item \begin{enumerate}
\item Let $Y$ be a set of 5 distinct points in $\P^2$. Let $V$ be the linear system of conics that contain $Y$. Show that $\dim(V) > 0$ if and only if at least four of the points are collinear.\footnote{Hint: Remember that $\dim(V) = 0$ if and only if there is \textit{exactly} one conic through those points.}

\item Let $Z$ be a set of 10 distinct points in $\P^2$. Let $W$ be the (possibly empty!) linear system of cubics that contain $Z$. Show that $\dim(W) > 0$ if and only if at least 6 of the points are collinear or at least 9 of the points lie on a conic.

\end{enumerate}

\item Let $F$ be an irreducible plane curve of degree $d$. Assume the partial derivative $F_x \neq 0$.

\begin{enumerate}
\item If $P$ is a point on $F$, show that $m_P(F_x) \geq m_P(F)-1.$
\item Using part (a) along with B\'{e}zout's theorem, show that $$\sum_{P \in V(F)} m_P (m_P -1) \leq d(d-1).$$ 
\item Conclude that $F$ has at most $\frac{1}{2}d(d-1)$ multiple points.
\item Give an example to show that the bound in (c) is not sharp. That is, show that there is some $d$ such that $F$ cannot have $\frac{1}{2}d(d-1)$ multiple points.

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\item Let $n \geq 1$ and $d \geq 1$. Define a map $v_d : \P^n \to \P^N$ by 
$$v_d ([x_0 : x_1: \ldots: x_n]) = [M_0 : M_1: \ldots : M_N],$$ 
where the $M_i$ are all the degree $d$ monomials in $n+1$ variables. That is, 
$$M_0 = x_0^d, \,\, M_1 = x_0^{d-1}x_1,\,\, \ldots, \,\, M_N = x_n^d.$$
This map is called the \textit{\textbf{degree $d$ Veronese embedding}} of $\P^n$.

\begin{enumerate}
\item What is $N$? Your answer should be in terms of $d$ and $n$.

\item Show that $v_d$ is well-defined and injective.\footnote{In fact, it's an isomorphism onto its image. You don't need to prove this.}

\item Denote $V := v_d(\P^n)$, the image of $\P(n)$. You may assume $V$ is Zariski closed.  Let $H \subset \P^N$ be a hyperplane. Show that $H \cap V = v_d(W)$, where $W \subset \P^n$ is a hypersurface of degree $d$.\footnote{$H \cap V$ is called a \textit{\textbf{hyperplane section}} of $V$.}

\item Show that the converse of (c) holds. Specifically, let $W \subset \P^n$ be a hypersurface of degree $d$. Show that there is some hyperplane $H \subset \P^N$ such that $H \cap V = v_d(W)$. That is, the hyperplane sections of $V$ are exactly the degree $d$ hypersurfaces in $\P^n$.

\item Notice that the twisted cubic in $\P^3$ is the degree 3 Veronese embedding of $\P^1$.\footnote{The Veronese embeddings of $\P^1$ are called \textit{\textbf{rational normal curves}}.} Apply the last sentence in (d) to the case of the twisted cubic. What does it say, concretely, in this case?


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