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\begin{document}


\begin{center}
  \Large{\textsf{\textbf{MATH 7141 - Algebraic Geometry I}}}\\
\vspace{1ex}
  \Large{\textsf{\textbf{Homework 7}}}\\
\vspace{2ex}

\large{\textbf{\textsf{Dimension theory, tangent spaces, smoothness}}}\normalsize

\vspace{1ex}

\end{center}
\normalsize

Please indicate any source in the literature used in finding the solution to a given problem. You are encouraged to work in teams. If you do so, please indicate the name of your collaborator(s) on your submission.
\vspace{1ex}

Solutions to each problem can be uploaded (on Carmen) by at most one student. There is no deadline, so work at your own pace.
\vspace{1ex}

Please use the following name for the file you upload \texttt{HW\#\_Problem\#.pdf}.
\vspace{3ex}




\hrule
\vspace{1ex}



\begin{center}
In all problems below, we assume $\KK$ is an arbitrary field, unless otherwise indicated. 
\end{center}

\vspace{1ex}

\hrule

\vspace{2ex}

In Lecture 30 we gave a definition of dimension of irreducible projective varieties $X$ in $\pr^n$ by means of linear projections from maximal linear subspaces of $\pr^n$ disjoint from $X$. The next two exercises involve such construction.\\

\pro{1} Let $X$ be the twisted cubic in $\pr^2_{\KK}$ defined in Problem 2 of Homework 2. Assume $\KK$ is algebraically closed.
\begin{enumerate}[(i)]
\item Find a linear subspace $\Lambda=\pr(W)$ in $\pr^2=\pr(\KK^3)$ of maximal dimension disjoint from $X$ and compute the linear projection $\pi\colon X \to Y=\pr(\KK^3/W)$, as defined in Lecture 30.
\item Show that $\pi$ is a finite morphism by explicitly computing the morphism \[
  \varphi = \pi^{\sharp}_{Y} \colon \cO_Y(Y)\to \cO_X(X).\]
  \end{enumerate}
\ \\
\pro{2} Let $X$ be the Veronese surface in $\pr^5$ over an algebraically closed field $\KK$.
\begin{enumerate}[(i)]
\item Find a linear subspace $\Lambda=\pr(W)$ in $\pr^2=\pr(\KK^3)$ of maximal dimension disjoint from $X$ and compute the linear projection $\pi\colon X \to Y=\pr(\KK^3/W)$, as defined in Lecture 30.
 \item Show that $\pi$ is a finite morphism by explicitly computing the morphism \[
  \varphi = \pi^{\sharp}_{Y} \colon \cO_Y(Y)\to \cO_X(X).\]
  \end{enumerate}
\ \\




\pro{3} Fix $X\subset \Aff^n$ an irreducible affine variety over  an algebraically closed field $\KK$. Consider $\overline{X} \subset \pr^n$, a linear subspace $\Lambda=\pr(W)$ in $\pr^2=\pr(\KK^{n+1})$ of maximal dimension disjoint from $\overline{X}$ and the linear projection $\pi\colon \overline{X} \to \pr(\KK^{n+1}/W)=\pr^r$, as defined in Lecture 30. Show that the restriction of $\pi$ to $X$ yields a finite surjective morphism $X\rightarrow \Aff^r$.\\


\pro{4} Given three irreducible affine varieties $X_1, X_2, X_3$  over an algebraically closed field $\KK$ with $X_1\subseteq X_2 \subseteq X_3$, show that
\[
\codim_{X_3} X_1 = \codim_{X_2} X_1 + \codim_{X_3} X_2.
\]
\vspace{3ex}

\pro{5} Let $X$ be a topological space and $U\subseteq X$ an open subset. Let $Y\subseteq X$ be an irreducible closed set with $U\cap Y\neq \emptyset$. Show that $U\cap Y$ is irreducible and closed in $U$ and, furthermore, we have
\[\codim_X(Y) = \codim_{U} (U\cap Y).
\]
\vspace{3ex}

\pro{6} Show that if $X$ and $Y$ are quasi-affine varieties, then
\[\dim(X\times Y) = \dim(X)+ \dim(Y).\]
\vspace{3ex}

\noindent Recall from Problem 10 Homework 6 the definition of locally closed sets.
\vspace{1ex}

\pro{7} Let $X$ be a quasi-affine variety and $Z$ a locally closed subset of $X$. Show that
\[\dim(Z) = \dim(\overline{Z}) > \dim(\overline{Z}\smallsetminus Z).
\]
\vspace{2ex}

\pro{8} Show that if $X$ and $Y$ are irreducible affine varieties of $\Aff^n$, then every irreducible component $Z$ of $X\cap Y$ satisfies:
\[\dim(Z)\geq \dim(X) + \dim(Y) - n.
\]
(\emph{Hint:} Describe $X\cap Y$ as the intersection of $X\times Y\subset \Aff^n\times \Aff^n$ with the diagonal $\Delta = \{(x,x) : x\in \Aff^n\}$.)\\

\noindent Given an abstract variety $X$ over an algebraically closed field $\KK$,  and a point $p$ in $X$, we define  the \emph{local dimension} of $X$ at $p$ as the quantity:
\[\dim_p(X) := \dim \cO_{X,p}.\]


\pro{9} Let $X$ be an abstract variety over an algebraically closed field $\KK$,  and fix a point $p$ in $X$. If $X=X_1\cup\ldots \cup X_s$ is the irreducible decomposition of $X$, show that
\[\dim_p X = \max_{1\leq i\leq s} \{\dim_p X_i : p\in X_i\}.
\]
(\emph{Hint:} Reduce the computation to the case when $X$ is affine, by working with an affine open cover of $X$. Prove the statement for the affine case, by identifying a chain of prime ideals in $\cO_{X,p}$ with a chain of prime ideals in $\KK[X]$.)\\

\noindent For the following exercise,  use the results from Problems 11 and 12 in Homework 2:
\vspace{1ex}

\pro{10} Find $\dim V(I)$ for the following monomial ideals:
\begin{enumerate}[(i)]
\item  $I=\langle xy^3z, x^2\rangle \subseteq \CC[x,y,z]$;
\item   $I=\langle wx^2z, w^3y, wxyz, x^5y^4\rangle \subseteq \CC[x,y,z,w]$.\\
\end{enumerate}

\pro{11} Let $I\subset \KK[x_1,\ldots, x_n]$ be a monomial ideal, where $\KK$ is an arbitrary closed field. Show that $\dim(V(I))=0$ if and only if for each $i\in \{1,\ldots, n\}$ we can find $m_i\geq 1$ with $x_i^{m_i}\in I$. \\


\pro{12} Fix $X\subseteq \pr^n$ a projective variety over an algebraically closed field $\KK$, and let $p\in X$ be any point. Pick $i\in \{0,\ldots, n\}$ such that $p$ lies in the affine patch $U_i$. Show that the projective tangent space $T_pX\subseteq \pr^n$ (which contains the point $p$) satisfies:
\[
(T_pX )\cap U_i = T_{\overline{p}}(X\cap U_i) + {\overline{p}}.
\]
Here,  $\overline{p}$ corresponds to the image of $p$ under the isomorphisms $U_i\simeq \Aff^n$, and $T_{\overline{p}}(X\cap U_i)$ denotes the tangent space of the affine variety $X\cap U_i\subseteq \Aff^n$ at the point $\overline{p}$. (\emph{Hint:} you may assume $p\in U_0$ up to reordering the coordinates of $\pr^n$.)\\



\pro{13} Given two varieties $X$ and $Y$ and points $x\in X, y\in Y$, the projection maps $\pi_1\colon X\times Y\to X$ and $\pi_2\colon X\times Y\to Y$ induce a linear map:
\[
T_{(x,y)} (X\times Y) \to T_xX \times T_y Y.
\]
Write down this map explicitly and show that it is an isomorphism.\\

\pro{14} Find all singular points of the following affine varieties in $\Aff_{\KK}^^n$ where $\KK$ is an algebraically closed field:
\begin{enumerate}[(i)]
  \item $X= V(y^2-x^3+3) \subseteq \KK^2$
\item $X=V(x^2y+x^3+y^3)\subseteq \KK^3$\\
\end{enumerate}

\pro{15} Find all singular points of the following affine varieties in $\Aff_{\KK}^^n$ where $\KK$ is an algebraically closed field:
\begin{enumerate}[(i)]
  \item $X= V(x^2y^2+x^2+y^2+2xy(x+y+1)) \subseteq \KK^2$,
\item $X=V(x^3-zxy+y^3)\subseteq \KK^3$
\end{enumerate}



\end{document}
