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\begin{document}
\begin{center}
{Homework Set 3}
\end{center}
\bigskip
1. (a) Prove that if $X$ is affine over $S$ then every $S$-morphism
$\PP ^{n}_{S}\rightarrow X$ factors as $s\circ p$, where $s$ is an
$S$-morphism from $S$ to $X$ (that is, a section of $X\rightarrow S$),
and $p\colon \PP ^{n}_{S}\rightarrow S$ is the structure morphism for
$\PP ^{n}_{S}$ as a scheme over $S$. In particular, if $S = \Spec
(k)$, where $k$ is a field, then every $k$-morphism from $\PP
^{n}_{k}$ to an affine $k$-scheme $X$ is constant, {\it i.e.}, it
factors through the reduced one-point scheme $\Spec (k)$.
\smallskip
(b) Deduce that if $k$ is a commutative ring (not the zero ring) and
$n>0$, then a vector bundle over $\PP ^{n}_{k}$ cannot be an affine scheme.
\smallskip
(c) Construct an example of an affine variety $X$ and a morphism $\pi
\colon X\rightarrow \PP ^{n}_{k}$ for some $n>0$ such that $X$ is an
affine line bundle over $\PP ^{n}_{k}$, {\it i.e.}, $\PP ^{n}_{k}$ can
be covered by open sets $U$ such that $\pi ^{-1}(U)$ is isomorphic to
$\Aa ^{1}_{U}$ as a scheme over $U$. Why does this not contradict
part (b)?
\smallskip
[The hint I orginally gave on (c), to construct $X$ as a surface
$V(f)$ in $SL_{2}(k)$, and map it to $\PP ^{1}_{k}$ by acting with
$SL_{2}(k)$ on a point $p\in \PP ^{1}_{k}$, was not helpful.
A better hint is to show that $\PP ^{1}\times \PP ^{1}$ minus the
diagonal is an affine variety $X$. In fact, $X$ is then the quotient
$SL_{2}(k)/T$, where $T$ is the subgroup of diagonal matrices in
$SL_{2}(k)$, and the projections to $\PP ^{1}$ coincide with the
action of $SL_{2}(k)$ on either of the two points $p\in \PP ^{1}$ that
are fixed by $T$. It is also true that $X$ is isomorphic to a surface
in $SL_{2}(k)$, but it does not embed in $SL_{2}(k)$ as a section of
$SL_{2}(k)\rightarrow SL_{2}(k)/T = X$.]
\medskip
2. Let $X$ be a scheme and let $\Fcal $ be a quasi-coherent sheaf on
$X$ which is locally finitely generated. Let $S = S(\Fcal )$ be the
symmetric algebra of $\Fcal $ over $\Ocal _{X}$, and let $V = \Spec
(S)$, a scheme affine over $X$. (In the case where $\Fcal $ is
locally free, $V$ is the geometric vector bundle whose sheaf of
sections is dual to $\Fcal $.) Note that $V$ has a distinguished
'zero section' over $X$, corresponding to the surjective homomorphism
$S\rightarrow S/S\Fcal \cong \Ocal _{X} $.
\smallskip
(a) Let $U\subseteq V$ be the open subset complementary to the zero
section. Show that the image of $U\rightarrow X$ is equal to the
support of $\Fcal $; in particular it is closed.
\smallskip
(b) Find an example showing that if $\Fcal $ is not assumed to be
locally finitely generated, then the image of $U$ is not necessarily
closed, and is not necessarily equal to the support of $\Fcal $. Such
an example will also show that the image of $\Proj (S)\rightarrow X$
need not be closed when $S$ is a quasi-coherent graded $\Ocal _{X}$
algebra that is not locally finitely generated.
\medskip
3. Recall that any invertible sheaf $\Lcal $ on a scheme $X$ gives
rise to a canonical morphism $W\rightarrow \Proj (\Gamma _{+}(\Lcal
))$, where $W$ is an open subset of $X$, the union of the sets $X_{f}$
for $f\in \Lcal (X)$, and $\Gamma _{+}(\Lcal )$ is the graded algebra
$\bigoplus _{d\geq 0} \Lcal ^{\otimes d}(X)$.
Let $X$ be the non-separated gluing of two copies of $Y = \Aa ^{1}_{k}
= \Spec k[x]$ ($k$ a field) along the open set $Y_{x}$.
\smallskip
(a) Classify the invertible sheaves $\Lcal $ on $X$, up to
isomorphism. Which ones are generated by their global sections?
\smallskip
(b) For each $\Lcal $ describe explicitly the open set $W$ and the
morphism $W\rightarrow \Proj (\Gamma _{+}(\Lcal ))$.
\medskip
4. Show that every {\it degree-2 hypersurface} $V(f)\in \PP ^{3}_{\CC
}$, where $f$ is a homogeneous quadratic polynomial in 4 variables, is
isomorphic to one of the following:
(i) A non-reduced scheme $X$ such that $X_{\red }\cong \PP ^{2}_{\CC }$,
(ii) A union of two projective planes $\PP ^{2}(\CC )$ intersecting
along a line $\PP ^{1}(\CC )$,
(ii) The projective closure of the cone $z^{2} = xy$ in $\Aa ^{3}$, or
(iii) $\PP ^{1}_{\CC } \times \PP ^{1}_{\CC }$.
To what extent does this classification depend on the ground field
being the complex numbers?
\medskip
5. Let $X = \Spec (\CC [z^{\pm 1}])$, that is, $X$ is the scheme such
that $X_{\cl } = \CC ^{\times }$ as a classical variety over $\CC $.
Setting $Y = X$, let $f\colon X\rightarrow Y$ be the morphism given by
$z\mapsto z^{2}$. The morphism $z\rightarrow -z$ then generates an
action of the cyclic group $G$ of order $2$ on $X$ by automorphisms as
a scheme over $Y$.
In the analytic topology on $\CC ^{\times }$, $X$ is a principal $G$
bundle over $Y$, that is, we can cover $Y = \CC ^{\times }$ by open
sets $U$ such that $f^{-1}(U)\subseteq X$ is isomorphic to $G\times U$
as a topological space (and also as a complex analytic manifold)
equipped with an action of $G$ by automorphisms over $U$.
\smallskip
(a) Show that we can identify $G$ with the underlying set of an affine
group scheme over $\CC $ in such a way that it acts algebraically on
$X$. (More generally one can do this for any finite group acting by
automorphisms of an algebraic variety.)
\smallskip
(b) Show that the action of the group scheme $G$ on the fiber
$f^{-1}(y)$ over each closed point of $y$ is isomorphic to the action
of $G$ on itself by left multiplication.
\smallskip
(c) Show that $X$ is not a principal $G$ bundle over $Y$ in the
Zariski topology.
\medskip
6. The set of pairs $(A,B)$ of commuting $n\times n$ matrices, over
an algebraically closed field $k$, is an affine algebraic variety $X$,
defined by obvious equations. By an old theorem of Motzkin and
Taussky, $X$ is irreducible.
\smallskip
(a) Given the Motzkin-Taussky theorem, find the dimension of $X$.
\smallskip
(b) Use this to prove that for every $n\times n$ matrix $A$, the space
of matrices that commute with $A$ has dimension at least $n$ (you can
do this without using Motzkin-Taussky by considering a suitable
irreducible component of $X$).
\medskip
7. Let $G(n,k)$ denote the Grassmann variety (over $\CC $) of vector
subspaces $V\subseteq \CC ^{n}$ of dimension $\dim (V) = k$. In class
we used the theorem that projective morphisms are proper to prove that
the set of pairs $(V,W)\in G(n,k)\times G(n,l)$ such that $\dim (V\cap
W)\geq m$ is a closed subvariety $X$.
Make this more explicit by proving that $X$ is irreducible (assuming
that $m\leq k,l\leq n$, so $X$ is non-empty), finding the dimension of
$X$, and finding homogeneous equations in the Pl\"ucker coordinates on
$G(n,k)\times G(n,l)$ whose zero locus is $X$. For a real challenge,
prove that your equations actually generate the ideal of $X$ in the
homogeneous coordinate ring of $G(n,k)\times G(n,l)$.
\medskip
8. The degree $d$ {\em Veronese map} $\PP _{k} ^{1}\rightarrow
\PP_{k} ^{d}$ is given by $(x:y)\mapsto (x^{d}:x^{d-1}y:\cdots
:xy^{d-1}:y^{d})$.
\smallskip
(a) Show that the Veronese map is the same as the projective embedding
given by the ample line bundle $\Lcal = \Ocal (d)$ on $\PP _{k}^{1}$
and a basis of its $k$-module of global sections.
\smallskip
(b) Assuming for simplicity that $k$ is an algebraically closed field,
show that the image of the degree 3 Veronese map, considered as a
reduced closed subscheme of $\PP_{k} ^{3}$, is given by $V(I)$ for the
graded ideal $I = (u^{2}-t v, v^{2}-u w, t w - u v)$, in coordinates
$(t:u:v:w)$ on $\PP ^{3}$. If you omit the last equation, taking $J =
(u^{2}-t v, v^{2}-u w)$, how does $V(J)$ differ (if at all) from
$V(I)$?
\medskip
9. Let $f$ be a homogeneous polynomial of degree $d$ in $n+1$
variables over a field $k$, so $X = V(f)\subseteq \PP _{k}^{n}$ is a
{\em degree $d$ hypersurface}. Compute $H^{i}(X,\Ocal (m))$ for all
$i$ and $m$. (These cohomology groups are finite dimensional vector
spaces over $k$, whose dimensions depend only on $n$, $d$, $m$ and
$i$, and not on the specific choice of $f$.)
\end{document}