\magnification\magstep1
\nopagenumbers

%%%%%%%%%%%Blackboardbold 
\font\gbbb=msbm10 scaled \magstep1
\font\bbbf=msbm10 
\font\sbbb=msbm6 
\font\ssbbb=msbm5 
\textfont6=\bbbf
\scriptfont6=\sbbb 
\scriptscriptfont6=\ssbbb 
\def\bbb{\fam6}
\def\mP{{\bbb P}} 
\def\mA{{\bbb A}} 
\def\mB{{\bbb B}} 
\def\mR{{\bbb R}}
\def\mZ{{\bbb Z}}

%%%%Gothic
\font\ggothic=eufm10 scaled \magstep1
\font\gothicf=eufm10
\font\sgothic=eufm7
\font\ssgothic=eufm5
\textfont5=\gothicf
\scriptfont5=\sgothic
\scriptscriptfont5=\ssgothic
\def\gothic{\fam5}


\font\Kopfont=cmbx12
\def\mapright#1{\smash{\mathop{\longrightarrow}\limits^{#1}}}
\def\mapdown#1{\Big\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\downmap#1{\downarrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\mapup#1{\Big\uparrow\rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}}
\def\longlongrightarrow{\relbar \joinrel \longrightarrow}
\def\cC{{\cal C}}
\def\cD{{\cal D}}
\def\gp{{\gothic p}}
\def\gq{{\gothic q}}
\def\Spec{\mathop{\rm Spec}}
\def\Proj{\mathop{\rm Proj}}

\centerline{\Kopfont Schemes}

\smallskip
\centerline{Exercises 3}

\bigskip\noindent
{\bf Schemes -- examples are important}

\bigskip\noindent
Please also argue that your examples are as required. Feel free to quote
results from Hartshorne or EGA.

\bigskip\item{\bf 1.} Give an example of a morphism of {\it integral}
schemes $f : X \to Y$ such that the induced maps ${\cal O}_{Y,f(x)}
\to {\cal O}_{X,x}$ are surjective for all $x\in X$, but $f$
is not a closed immersion.

\medskip\item{\bf 2.} Give examples of graded rings $S$ such that
\itemitem{\bf (a)} $\Proj(S)$ is affine and nonempty, and
\itemitem{\bf (b)} $\Proj(S)$ is integral, nonempty but not isomorphic
to ${\bbb P}^n_A$ for any $n\geq 0$, any ring $A$.

\medskip\item{\bf 3.} Give an example of a nonconstant morphism
of schemes ${\bbb P}^1_{\bbb C} \to {\bbb P}^5_{\bbb C}$ over
$\Spec({\bbb C})$.

\medskip\item{\bf 4.} Give an example of an isomorphism of schemes
${\bbb P}^1_{\bbb C} \to \Proj({\bbb C}[X_0,X_1,X_2]/(X_0^2+X_1^2+X_2^2))$.

\medskip\item{\bf 5.} Give an example of a morphism of schemes
$f : X \to {\bbb A}^1_{\bbb C}=\Spec({\bbb C}[T])$ such that the
(scheme theoretic) fibre of $f$ over $t \in {\bbb A}^1_{\bbb C}$ is (a)
isomorphic to ${\bbb P}^1_{\bbb C}$ when $t$ is a closed point not equal
to $0$, and (b) not isomorphic to ${\bbb P}^1_{\bbb C}$ when $t=0$. 

\medskip\noindent
{\bf Remark.} This can be done in many, many ways. Here are some additional restraints
you can impose: Can you do it with fibre at $t=0$ projective? Can you do it
with special fibre irreducible and projective? Can you do it with special
fibre integral and projective? Can you do it with fibre at $t=0$ smooth and
projective? What about similar questions when you replace
${\bbb P}^1_{\bbb C}$ with another variety over ${\bbb C}$?

\bye
