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\begin{document}
\begin{center}
{Homework Set 1---Classical varieties}
\end{center}
\bigskip
The problems on this set deal with varieties over an algebraically
closed field $k$. For these problems you can work with Serre's
definition of a classical variety, as we did in the examples at the
beginning of the semester.
Alternatively, you can interpret the problems as questions about
schemes, working with the scheme $\Spec (\Ocal (X))$ for a classical
affine variety $X$, or a gluing of affine schemes for a general
variety covered by affine open subsets. For example, the scheme $\PP
^{n}_{k}$ corresponds to classical projective space $\PP ^{n}(k)$.
In class we already mentioned that classical varieties are in a
certain sense equivalent to reduced schemes locally of finite type
over $k$. Eventually we will describe the equivalence in full detail.
This equivalence implies that a solution to any of the problems for
varieties also solves the scheme version, and vice versa.
Nevertheless, you may find it instructive to think about how to solve
some of the problems explicitly using varieties first, and then
schemes. Normally you should find that the solution involves the same
steps (constructing various ring homomorphisms, etc.) in either
setting. The scheme solution will often be more general: for
instance, it may solve a version of the problem in which the schemes
involved are defined over any ring $R$, not just an algebraically
closed field $k$.
\medskip
Extra challenging problems are marked with (*).
\bigskip
1. In class we saw that the parametrization $t\rightarrow
(t^{2},t^{3})$ of the curve $C=V(y^{2}-x^{3})\subseteq \Aa ^{2}$ is a
morphism $\phi \colon \Aa ^{1}\rightarrow C$ which is a homeomorphism,
but not an isomorphism of varieties. This doesn't rule out the
possibility that $C$ is isomorphic to $\Aa ^{1}$ via some other
morphism. Show that in fact $C$ is not isomorphic to $\Aa ^{1}$, by
proving that the subring $k[t^{2},t^{3}]$ of $k[t]$ is not isomorphic
as a $k$-algebra to $k[t]$.
\medskip
2. Let $C$ be the closure in $\PP ^{2}$ of the plane curve
$V(x^{2}+y^{2}-1)$, that is, $C = V(x^{2}+y^{2}-z^{2})$ in projective
coordinates, where $V(z)$ is the line at infinity, and $U_{z}$ is
identified with the affine $(x,y)$-plane. Let $L_{t}$ be the
projective closure of the line $V(y-t(x+1))$, {\it i.e.}, $L_{t}$ is
any line through $(-1,0)$ except for the vertical line $x = -1$.
\smallskip
(a) Show that $L_{t}$ meets $C$ in exactly one point $Q_{t}$ other
than $(-1,0)$.
\smallskip
(b) Identifying the set of all lines through $(-1,0)$ with $\PP ^{1}$,
show that the map sending $L_{t}$ to $Q_{t}$ extends uniquely to an
isomorphism $\PP ^{1}\rightarrow C$. Where does the vertical line $L
= V(x+1)$ go?
\smallskip
(c) Assuming $\ch (k)\not =2$, show that $C$ has two points on the
line at infinity $V(z)$, and that these correspond under the
isomoprhism in (b) to the lines $L_{t}$ with $t = \pm \sqrt{-1}$.
What happens if $\ch (k) = 2$?
\smallskip
(d) Prove that if $k$ is any field, not necessarily algebraically
closed, the answer to (a) gives a parametrization of all solutions of
the equation $x^{2}+y^{2}=1$ in $k^{2}$, excluding $(-1,0)$, by
elements $t\in k$ such that $t^{2}\not =-1$. Use this to show that
every triple of positive integers $a^{2}+b^{2}=c^{2}$ with no common
factor has the form $(p^{2}-q^{2}, 2pq, p^{2}+q^{2})$ for integers
$p>q>0$.
\medskip
3. (a) Show that the set of curves of degree $d$ in in $\PP ^{2}$ is
naturally parametrized by $\PP ^{n}$, where $n = \binom{d+2}{2}-1$.
Let $H\subseteq \PP ^{n}\times \PP ^{2}$ be the `tautological family'
whose fiber over a point of $\PP ^{n}$ is the curve parametrized by
that point. Note that if $f$ is a polynomial separately homogeneous
in the projective coordinates on $\PP ^{n}$ and $\PP ^{2}$, then the
zero locus $V(f)\subseteq \PP ^{n}\times \PP ^{2}$ makes sense and is
a closed subvariety. Find such an $f$ so that $H = V(f)$.
\smallskip
(b) For $d=2$, let $Y$ be the locus in $\PP ^{5}$ which parametrizes
quadratic curves (conics) that degenerate to two lines or a double
line. Show that $Y$ is a closed subvariety of $\PP ^{5}$ and find its
equation(s).
\smallskip
(c*) Repeat (b) for $d=3$, where $Y\subseteq \PP ^{9}$ parametrizes
cubic curves that degenerate to a line and a conic, or three lines, or
a line and a double line, or a triple line.
\smallskip
Remark: for general $d$, the degenerate curves, {\it i.e.}, those that
are not reduced and irreducible, are parametrized by a closed
subvariety $Y\subseteq \PP ^{n}$. This follows from `elimination
theory,' which is another way of saying that it is a corollary to the
geometric theorem that {\it projective morphisms are proper}. We'll
eventually prove the latter theorem.
\medskip
4. We can regard $\operatorname{SL}_{2}(k)$ as the affine variety
$V(ad-bc-1)$ in the space $\Aa ^{4}(k)$ of $2\times 2$ matrices
$\begin{pmatrix} a& b\\ c& d \end{pmatrix}$.
\smallskip
(a) Verify that the group law in $\operatorname{SL}_{2}(k)$ and the
map sending an element to its inverse are morphisms of algebraic
varieties.
\smallskip
(b) Identifying $\PP ^{1}(k)$ with the set of one-dimensional
subspaces of $k^{2}$, the group $\operatorname{SL}_{2}(k)$ acts on it,
via its action on $k^{2}$. More explicitly, an element of
$\operatorname{SL}_{2}(k)$ sends $(z:w)\in \PP ^{1}$ to $(z':w')$,
where
\[
\begin{pmatrix} z'\\
w' \end{pmatrix} = \begin{pmatrix} a& b\\
c& d \end{pmatrix} \begin{pmatrix} z\\
w \end{pmatrix}.
\]
Show that on the affine line $\{(z:1) \}\subset \PP ^{1}$, this action
is the fractional linear transformation sending $z$ to
$(az+b)/(cz+d)$, for $cz+d\not =0$.
\smallskip
(c) Show that the action of any fixed matrix in
$\operatorname{SL}_{2}(k)$ is a morphism from $\PP ^{1}$ to $\PP
^{1}$, by covering $\PP ^{1}$ with affine open sets on which it is
given by a polynomial map (you will need different open coverings on
$\PP ^{1}$ regarded as the domain and codomain of the map).
\smallskip
(d*) Show that the group action as a whole is a morphism
$\operatorname{SL}_{2}(k)\times \PP ^{1}\rightarrow \PP ^{1}$.
\smallskip
(e*) Show that the group $\operatorname{PSL}_{2}(k) =
\operatorname{SL}_{2}(k)/\{ \pm I\}$ is also an affine variety, in
such a way that the group law and the action of
$\operatorname{PSL}_{2}(k)$ on $\PP ^{1}$ are morphisms. Hint: the
coordinate ring of $\Ocal (\operatorname{PSL}_{2}(k))$ is the subring
of $\Ocal (\operatorname{SL}_{2}(k))$ consisting of functions constant
on cosets of $\{\pm I \}$.
\medskip
5. Prove that every global regular function on projective space $\PP
^{n}$ is constant. If you do this using varieties, you should assume
that the ring of global regular functions $\Ocal_{X} (X)$ on a
classical affine variety $X$ is equal to the coordinate ring $\Ocal
(X)$. This is a corollary to the corresponding theorem for schemes,
which we already proved, and the equivalence relating the classical
variety $X$ to the scheme $\Spec (\Ocal (X))$.
\medskip
6. (a) Let $X$ be the hypersurface $X=V(xz-wy)\subseteq \Aa ^{4}$.
Let $U= X_{y}\cup X_{z} = X-V(y,z)$. Find a regular function $g\in
\Ocal _{X}(U)$ which cannot be expressed in the form $g=h/f$, where
$h$ and $f$ are polynomials in the coordinates $w,x,y,z$ such that
$f\not =0$ on $U$.
\smallskip
(b) Prove that $U$ is isomorphic to the complement of a line in $\Aa
^{3}$. In particular, $U$ is not affine. Hint: use the function $g$
as one of the coordinates on $\Aa ^{3}$.
\smallskip
(c*) The phenomenon in (a) can occur even if $X$ and the open subset
$U$ are both affine. To see this, let $X = V(xz-wy, a y^{2}+b z^{2}-y
z)\subset \Aa ^{6}$ and let $U=X_{y}\cup X_{z}$. Prove that in this
case, $U$ is isomorphic to the affine variety $V(u y+ v z-1)\subseteq
\Aa ^{5}$, with the open embedding $U\hookrightarrow X$ corresponding
to a ring homomorphism $y\mapsto y$, $z\mapsto z$, $x\mapsto g y$,
$w\mapsto g z$, $a\mapsto u z$, $b\mapsto v y$, where $u,v,y,z,g$ are
the coordinates on $\Aa ^{5}$.
\medskip
7. Let $(x_{0}:\cdots :x_{3})$ be projective coordinates on $\PP
^{3}$. Let $U_{0},\ldots,U_{3}$ be the standard affines $U_{i} = \PP
^{3}-V(x_{i})$, with coordinates $\{x_{j}\mid j\not =i \}$ on $U_{i}$
given by fixing $x_{i}=1$.
\smallskip
(a) Show that there is a projective variety $Z\subseteq \PP ^{3}$ such
that $Z\cap U_{i}$ is given by the equations
\begin{align*}
x_{2}=x_{1}^{2},\; x_{3}=x_{1}^{3}&\quad \text{on $U_{0}$}\\
x_{0}x_{2}=1,\; x_{3}=x_{2}^{2}&\quad \text{on $U_{1}$}\\
x_{1}x_{3}=1,\; x_{0}=x_{1}^{2}&\quad \text{on $U_{2}$}\\
x_{1}=x_{2}^{2},\; x_{0}=x_{2}^{3}&\quad \text{on $U_{3}$}.
\end{align*}
(b) Find homogeneous equations of $Z$ in projective coordinates.
\smallskip
(c) Construct a morphism $\PP ^{1}\rightarrow \PP ^{3}$ whose image is
$Z$. Is $Z$ isomorphic to $\PP ^{1}$?
\medskip
8. (a) Prove that for every non-zero linear form $f =
a_{0}x_{0}+\cdots +a_{n}x_{n}$ in the projective coordinates $(x_{0}:
\cdots :x_{n})$, the open subvariety $U=\PP ^{n}-V(f)$ is isomorphic
to $\Aa ^{n}$.
\smallskip
(b*) Prove that for every non-zero homogeneous polynomial
$f(x_{0},\ldots,x_{n})$, the open subvariety $U=\PP ^{n}-V(f)$ is
affine. Hint: for coordinates on $U$ take all the functions
$x^{m}/f$, where $x^{m}$ is a monomial in the $x_{i}$ of degree $d =
\deg (f)$. Then show that $U$ is isomorphic to the affine variety $X$
with coordinate ring $\Ocal (X) = R/(f(x)-1)$ where $R\subseteq
k[x_{0},\ldots,x_{n}]$ is the subalgebra generated by monomials of
degree $d$.
\medskip
9. We can identify the graph of a polynomial function
$f(x_{1},\ldots,x_{n})$ with the affine variety $X =
V(y-f(x))\subseteq \Aa ^{n+1}$. Prove that $y-f(x)$ generates the
ideal $\Ical (X)\subseteq k[x_{1},\ldots,x_{n},y]$ and that $X$ is
isomorphic to $\Aa ^{n}$.
\medskip
10. Let $X = V(y-f(x))\subseteq \Aa ^{n+1}$ be the graph of $f$, as
in the previous problem. Construct a natural bijective correspondence
between ring homomorphisms $\Ocal (X)\rightarrow k[t]/(t^{2})$ and
pairs $(p,v)$, where $p\in X$ and $v$ is a tangent vector to $X$ at
$p$, that is, a vector in $k^{n+1}$ such that the directional
derivative of $y-f(x)$ in the $v$ direction vanishes at $p$. Note
that the derivatives of a polynomial make sense formally with
coefficients in any field.
\end{document}