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\begin{document}
\pagestyle{plain}
\title{Intro to Algebraic Geometry, Problem Set 6}
\date{October 21, 1999.}
\maketitle

{\parskip=12pt % closing bracket is just before the bibliography 

Due Thursday October 28 in class (no lates).
Hand in seven of the following questions.
You're strongly encouraged to collaborate
(although write up solutions separately), and you're also
strongly encouraged to ask me questions (if you're stuck, or if the
question is vaguely worded, or if you want to try out an argument).
Ask someone else about the non-scheme problems you skip.

The ``miscellaneous'' questions are more straightforward.   The problems
on the Hilbert polynomial and Veronese embeddings are harder, but there are
fast shortcuts (which aren't necessarily easy to find). 

{\bf Miscellaneous.}
\begin{enumerate}
\item  (a)  Consider two varieties, $\proj^1$ and $\A^1$.  Let one of the standard affine opens of $\proj^1$ have
coordinate $t$, and let $x$ be a coordinate on $\A^1$.  Find a
dominant rational map $f: \proj^1 \dashrightarrow \A^1$ corresponding
to the morphism of function fields $k(\A^1) = \kb(x) \rightarrow
k(\proj^1) = \kb(t)$ given by $x \rightarrow t^2$.

\item[] (b) If $X$ is a prevariety, give a natural identification between elements of $k(X)$ and the 
rational maps from $X$ to $\proj^1$. 

\item (a) In Problem Set 4, you showed that the function field of the hypersurface $X$ given by $wx=yz$ in $\A^4$
was isomorphic to $\kb(t_1, t_2, t_3)$.  By comments in class, $X$
should be rational, i.e. birational to $\A^3$.  Prove that it is.

\item[]  (b) If $X$ is the line with the doubled origin, show that the diagonal $\De(X)$
is not closed in $X \times X$. 

\item Consider the morphism $f: \A^2 \setminus \{ (0,0 ) \} \rightarrow \proj^1$ given by $(x,y) \mapsto (x ;y)$.
Show that there is no morphism $\A^2 \rightarrow \proj^1$ extending
$f$.  

\end{enumerate}

{\bf Projective varieties: Hilbert functions, dimension, and degree.}
Suppose $V$ is a projective variety in $\proj^n$ (with projective
coordinates $x_0$, \dots, $x_n$) with projective coordinate ring $R =
\kb[x_0, \dots, x_n]/I = \oplus_d R_d$, where $R_d$ is the degree $d$
part of $R$.  It is a fact that there is a polynomial $f(d)$ and an
integer $d_0$ such that $\dim R_d = f(d)$ for $d \geq d_0$ (see Lang's
Algebra, the section on the Hilbert polynomial).  $f(d)$ is called the
{\em Hilbert polynomial} of $V \subset \proj^n$.  The {\em dimension}
of $V$ can be defined as the degree of $f(d)$ (although a proof is
required to show that is the same as the definition I'll give in the
course).  The {\em degree} of $V$ can be defined as $(\dim V)!$
times the leading coefficient of $f(d)$.   In the next five problems,
use these definitions of dimension and degree.  (Another, more intuitive,
definition of degree is:  Intersect $V$ with $\dim V$ randomly chosen hyperplanes,
and count the points of intersection.) 

\begin{enumerate}
\item[4.] Calculate the Hilbert polynomial when $V = \proj^n$. Verify that
$\dim V = n$ and $\deg V = 1$.   If $V$ is a codimension $m$ {\em linear subspace}, 
i.e. cut out by $m$ linear independent linear forms in the $x_i$, verify
that $\dim V = n-m$ and $\deg V = 1$.   

\item[5.] Suppose $V$ is cut out by a single equation $q(x_0, \dots, x_n) = 0$,
where $q$ is irreducible and homogeneous of degree $e$, so $R =
\kb[x_0, \dots, x_n]/(q)$.  Calculate the Hilbert polynomial of $V \subset \proj^n$,
and verify that $\dim V = n-1$ and $\deg V = e$.

\item[6.]  {\em (The beginnings of intersection theory.)}  
Suppose $V' \subset \proj^n$ is a projective variety of dimension $D$
and degree $d$, with coordinate ring $R= \kb[x_0, \dots, x_n]/I(V')$,
and suppose that $q$ (as in problem 5) cuts out an irreducible subvariety
$W$ of $V'$ (distinct from $V'$), which is equivalent to $q$ being
irreducible in the ring $R$.  Prove that $\dim W = D-1$ and $\deg W =
de$.

\end{enumerate}

{\em The degree $d$ Veronese embedding of $\proj^n$.}
Consider the morphism
$$
\proj^n \rightarrow \proj^{ \binom {n+d} d - 1}$$
given by $(x_0^d; x_0^{d-1} x_1; \dots; x_n^d)$, where 
$(x_0^d; x_0^{d-1} x_1; \dots; x_n^d)$ includes all degree $d$ monomials in $x_0$, \dots, $x_n$.
\begin{enumerate}
\item[7.] Verify that it is a morphism, and an isomorphism onto its image $V$.
\item[8.] If $R = \oplus_{e \geq 0} R_e$ is the projective coordinate ring of $V$,
 verify that $R_e$ can be identified with the vector space of (homogeneous)
degree $de$ polynomials in the $x_i$.  Hence verify that $\dim V = n$ and
$\deg V = d^n$.
\end{enumerate}

{\bf Schemes:  nilpotents.}
\begin{enumerate}
\item[9.] (a) If $f \in R$ is nilpotent, describe $D(f)$ in $\Spec R$.

\item [] (b) A scheme $(X, \oh_X)$ is {\em reduced} if for every open set $U \subset X$,
the ring $\oh_X(U)$ has no nilpotent elements.
Show that $(X,\oh_X)$ is reduced if and only if for every $P$ in $X$,
the local ring $\oh_{X,P}$ has no nilpotent elements.  (Translation:  ``reducedness''
is a local property, in the sense that you can check it on points.)

\item[10.]  Explain intuitively how the morphisms from $\Spec \kb[t]/ (t^2)$ (which has only one point)
to $\A^2$ (over the field $\kb$) correspond to the data of a point of $\A^2$, and a tangent vector of $\A^2$.
Using this ``definition'', calculate the dimension of the tangent space to $\A^n$ at the origin; and
the dimension of the tangent space to the curve $y^2 = x^2+x^3$ in $\A^2$ at the origin.

\end{enumerate}

} % end of parskip; it started just before the introduction
\end{document}

