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{Mathematics 535 \hfill Algebraic Geometry \hfill Fall 2002}

\bigskip


\centerline{ Problem Set 2 (Last revised \number\month/\number\day/\number\year)}
\bigskip




\item{2.2}{Determine the ring of regular functions on $\bA^2 -\{(0,0)\}$}
\itemitem{Proof:}{(T. Pham)}
\medskip

\item{2.3}{Use 2.2 to show that $\bA^n -\{(0,0,\dots,0)\}$
is not isomorphic to an affine variety when $n\geq 2$. What is the
situation for $n=1$?}


\itemitem{Proof:}{(T. Pham)}

\medskip

\item{2.5}{Show that the number of monomials of degree $d$ in $n+1$ variables
is $n+d\choose n$.}

\itemitem{Proof:}{Consider $n+d$ boxes in a row. Each monomial 
$x_0^{d_0}\cdots x_n^{d_n}$ corresponds to an ordered partition of d, and hence
to a choice of d boxes to place a $+$ sign, each of the remaining boxes to be filled
by a $1$.  This places the monomials in bijection with the 
$n+d\choose n$ ways to place the $+$ signs, which equals the $n+d\choose d$ ways
to choose the $d$ boxes to contain a 1.}

\medskip

\item{2.8}{Show that the image $Y=\nu_d(X)\subset \bP^N$ of a projective variety
$X\subset \bP^n$ is isomorphic to X via the Veronese map.}

\itemitem{Proof:}{(S. Ganguly)}

\medskip

\item{2.9}{Use 2.8 to show that any projective variety is isomorphic to the
intersection of a Veronese variety and a linear subspace, and hence
is isomorphic to an intersection of quadrics.}

\itemitem{Proof:}{(S. Ganguly)}




\medskip



\item{2.10}{Let  $Y=\nu_d(X)\subset \bP^N$ be the Veronese image 
of a projective variety $X\subset \bP^n$. What is the relation of the homogeneous
coordinate rings? Show that the homogeneous coordinate ring is not an isomorphism
invariant of a projective variety, and construct varieties in a projective
space that are isomorphic but not projectively equivalent.}

\itemitem{Proof:}{Let $S(X)$ be the homogeneous coordinate ring of $X$ and
$S(\nu_d(X))$ the homogeneous coordinate ring of its Veronese image. 
There is clearly a map of rings $S(\nu_d(X))\to S(X)$ obtained
by sending the coordinate $Z_I$ corresponding to multi-index $(i_0,\ldots,i_n)$
to the image of $x_0^{i_0}\cdots x_n^{i_n}$ in $S(X)$.  This clearly gives
an injective ring homomorphism from $S(\nu_d(X))$ to $S(X)$. The image is easy
to describe as it is generated by the images of degree d monomials in the $x_i$
in $S(X)$. For example when $d=2, X=\bP^1$ we have $S(X)=K[x_0,x_1]$
and $S(\nu_2(\bP^1))=K[Z_0,Z_1,z_2]/(Z_0Z_2-Z_1^2) \simeq K[x_0^2,x_0x_1,x_1^2]$.
The second ring is not isomorphic to the first, since it is not
generated as a ring by two elements (if it were they would be 2 polynomials of
degree at most 2 in the $x_i$ which span the 3 dimensional vector space of
polynomials of degree 2). This also provides an example of two isomorphic
varieties in $\bP^2$, that is $\nu^2(\bP^1)$ and $\bP^1=\{[x_0,x_1,0]\}
\subset \bP^2$ which have non-isomorphic homogeneous coordinate rings,
and hence are not projectively equivalent.}

\medskip

\item{2.12}{(i) Let $L,M,N \subset \bP^3$ be any three pairwise disjoint lines.
Show that the union of the lines in $\bP^3$ meeting all three lines is projectively
equivalent to the Segre variety $\Sigma_{1,1}\subset \bP^3$ and that this union is
the unique Segre variety containing these lines.\hfil\break
(ii) More generally suppose that $L,M,N$ are three disjoint k--planes in
$\bP^{2k-1}$.  Show that the union of all lines meeting $L,M$ and $N$
is projectively equivalent to the Segre variety $\Sigma_{k-1,1} \subset \bP^{2k-1}$
and that this is the unique Segre variety containing $L,M, N$.  Try to find an analogous
description of the Segre varieties $\Sigma_{a,b}$ for $a,b \geq 2$.}

\itemitem{Proof:}{We do the general case.  Choosing bases for the 
first two k-planes we may assume after a projective transformation
that they are the k-planes
$x_1=\ldots =x_k=0$ and $x_{k+1}=\ldots=x_{2k}=0$. Choose a basis $b_i$ of the
third k-plane.
There exist $k \times k$ matrices $M_1,M_2$ such that if we apply the
projective transformation corresponding to the $2k\times 2k$ matrix with
$M_1,M_2$ down the diagonal the first two planes are preserved and the
basis of the third plane consists of the vectors $w_i$ having
1 in positions $i, k+i$ and zeros elsewhere (the
k-vectors formed by the first k entries of the $b_i$ are independent,
otherwise it intersects the first plane. Let $M_1$ be the $k\times k$ matrix
which is the inverse of the matrix formed by taking the first k entries of
$b_i$ as rows. Similarly for the last k entries and $M_2$.)
Consider the map
$\phi: [\lambda,\mu]\times [x_1,\ldots,x_k] \mapsto \lambda[x_1,\ldots,x_k,x_1,\ldots,x_{k}]
+\mu[x_1,\ldots,x_k,0,0,0,0].$  For a fixed point in $\bP^{k-1}$ this traces out
a line in $\bP^{2k-1}$ which intersects all three k-planes (take $\lambda=0$, then
$\mu=0$, and finally $\lambda=-\mu=1$). Further, every point in a line which meets all
three k-planes is in the image, since the condition that a combination of a
point in the first k-plane with a point of the third is in the second is clearly that
the first k-coordinates of each determine the same point up to scalar multiples.
The union of all lines meeting all 3 k-planes is projectively equivalent
to the image of $\phi$.  The formula for $\phi$ shows that this image is
equivalent to the Segre embedding $\Sigma_{1,k-1}$.  
For uniqueness, note that any Segre variety containing the three k-planes
is given by quadric equations, so its restriction to a line meeting these
three planes is a quadratic equation with three zeros, hence vanishes on
the line. Thus the Segre variety contains the union of all lines meeting the
three k-planes.  In general $\Sigma_{a,b}, a \geq b$ is projectively equivalent to
the union of
all b-planes meeting b+2 disjoint (a+1)-planes.
}


\medskip

\item{2.13}{Show that the twisted cubic in $\bP^3$ is the intersection of the 
Segre threefold
with a three plane in $\bP^5$.}
\itemitem{Proof:}{1. The Segre threefold is the set of 
$$[w_0,\ldots,w_5]=\{[xz_0,yz_0,xz_1,yz_1,xz_2,yz_2]\}$$
in $\bP^5$. Consider the plane $w_1-w_2=0=w_3-w_4$.  The intersection consists
of all $[w_0,w_1,w_1,w_4,w_4,w_5]$ such that $w_0w_4=w_1^2, w_1w_5=w_4^2,
w_0w_5=w_1w_4$,
which are just the equations for the twisted cubic in $w_0,w_1,w_4,w_5$ 
space.}

\itemitem{}




\medskip

\item{2.14}{Show that any line $\ell \subset \Sigma_{2,1}\subset \bP^5$
is contained in the image of a fiber of $\bP^2 \times \bP^1$ over $\bP^2$ or $\bP^1$.

\itemitem{Proof:}{(J. Burdges)}










\medskip

 
\item{2.15}{Show that the image of the diagonal $\Delta \subset \bP^n\times \bP^n$
under the Segre map is the Veronese variety $\nu_2(\bP^n)$ lying in a subspace
of $\bP^{n^2+2n}$.  Conclude that in general the diagonal
$\Delta_X\subset X \times X$ is a subvariety of the product variety, and
similarly for the diagonal in the n--fold product $X^n$.}

\itemitem{Proof:}{(J. Burdges)}








\medskip


\item{2.17}{Let $C \subset \bP^1\times\bP^1$ be the zero locus of
an irreducible bihomogeneous polynomial $F(X,Y)$ of bidegree $(1,2)$.
Show that the image of $C$ under the Segre map $\sigma:\bP^1\times\bP^1 \to \bP^3$
is a twisted cubic curve.}

\itemitem{Proof:}{(J. Tedor)}


\medskip

\item{2.18}{Let $C_{\alpha,\beta}$ be the curve of problem 1.29 (a so-called rational
quartic curve).  Show that $C_{\alpha,\beta}$ lies on the Segre 
surface $S$ which is the zero set of $Z_0Z_3-Z_1Z_2$ in $\bP^3$.  Show that $S$ is the unique quadric surface
containing $C_{\alpha,\beta}$, and that $C_{\alpha,\beta}$ is the zero set of
a bihomogeneous polynomial of bidegree $(1,3)$ on $S \simeq \bP^1\times \bP^1$.
Use this to give an alternate solution to problem 1.29.}

\itemitem{Proof:}{(B. Kennedy)}




\medskip
\item{2.19}{Use 2.18 to show that there is a continuous family of curves 
$C_{\alpha,\beta}$ not projectively equivalent to one another.}

\itemitem{Proof:}{(B. Kennedy)}

\medskip

\item{2.20}{(a) Let $X \subset \bP^n, Y\subset \bP^m$ be projective
varieties. Show that the image $\sigma(X \times Y) \subset \sigma(\bP^n\times\bP^m)\subset
\bP^{mn+m+n}$ is a projective variety.\hfil \break
(b) Show that (a) is true if we replace projective by quasi-projective throughout.}

\itemitem{Proof:}{If X and Y are the zero sets of collections $\{f_\alpha(X)\}$
and $\{g_\beta(Y) \}$ of homogeneous polynomials of degree d then $\sigma(X\times Y)$
is clearly the zero set of the bihomogeneous polynomials obtained by considering
the polynomials $f,g$ as polynomials of bidegree $(d,0)$ and $(0,d)$, hence
it is a subvariety of the Segre variety.  For part (b) suppose that X, Y are locally
closed. Let $p \in X, q \in Y$ and let $U,V$ be open sets containing $p,q$ 
which are complements of closed sets $Z\subset \bP^n,W\subset \bP^m$ respectively.
By the first part $\sigma(Z\times W)$ is projective, and $\sigma(U\times V)$ is
open  (it's the complement of the closed subset
$\sigma(Z\times \bP^m)\cup \sigma(\bP^n \times W)$ )
Since $X,Y$ are locally closed there exist closed subsets $A\subset \bP^n, 
B\subset \bP^m$ such that $X \cap U= A \cap U ,Y \cap V= B \cap V$. Then
$\sigma(X\times Y)\cap \sigma(U \times V) =\sigma(A\times B)\cap \sigma(U \times V)$,
which implies part (b).}
\medskip

\item{2.22}{Show that the Zariski topology on the product variety $X\times Y$
is not the product
of the Zariski topologies of $X,Y$.}

\itemitem{Proof:}{(J. Tedor)}



\medskip

\item{2.24}{Let $X\subset \bP^n$ be a projective variety and let $\phi: X \to \bP^m$ be any regular map. Show that the graph $\Gamma_\phi \subset X\times \bP^n \subset \bP^n\times \bP^m$
is a subvariety.}

\itemitem{Proof:}{The product map $\phi \times i:X \times \bP^m \to \bP^m\times \bP^m$
is regular since projection on each factor is regular.  Hence the inverse image
of the diagonal $\Delta \subset \bP^m \times \bP^m$ is closed ( see 2.15).
This inverse image is $\{(x,y)| y=\phi(x)\}$ which is just $\Gamma_\phi$.}

\medskip
                                
\item{2.26}{Let $\phi:X \to B, \eta:Y \to B$ be maps of
algebraic varieties. Show that the variety $X\times_B Y
=\{(x,y) \in X\times Y: \phi(x)=\eta(y)\}$ is in fact a fiber product of Y over B in the
category of algebraic varieties.}

\itemitem{Proof:}{ $X\times_B Y$ has the structure of a projective variety
since it is the inverse image of the diagonal $\Delta \subset B \times B$
under the regular map $\phi \times \eta$. Suppose that $\alpha:Z \to X,
\beta: Z \to Y$ are regular maps such that $\phi \circ \alpha =\eta \circ
\beta$ (that is, $\alpha, \beta$ are maps in the category of varieties over
$B$.)  The product map $\alpha \times \beta:Z \to X\times Y$ is regular.
Further the image is contained in the subset $X \times_B Y$, and by the properties of
products this is the unique regular map $\psi:Z \to X\times_B Y$ such that
$\pi_X\circ \psi = \alpha, \pi_Y\circ \psi = \beta$.}

\medskip

\item{2.28}{Let $X_{2,2}$ be an example of the  rational normal scroll, 
that is the image of
the map $\phi$ obtained by following the map ${\rm Id} \times \nu_2:\bP^1\times \bP^1 \to
\bP^1\times \bP^2$ with the Segre map $\sigma:\bP^1\times \bP^2\to\bP^5$.
Find the polynomials which define $X_{2,2}$ and show that the rational normal curve in
$\bP^4$ may be realized as the intersection of $X_{2,2}$ with a hyperplane in
$\bP^5$.}


\itemitem{Proof:}{(K. Dalili)}

\medskip

\item{2.29}{Consider the variety $Q \simeq 
\bP^1\times\bP^1 \subset \bP^3$ defined by
$Z_0Z_3-Z_1Z_2$, and let $L \subset Q$ be the line given by $Z_1=Z_0=0$.  Show that the
vector space of all homogeneous quadratic polynomials in $Z_i$ which vanish on
$L$ has dimension 7.  Show that the map $\phi$ of problem 2.28 may be given as
$\phi([Z_0,Z_1,Z_2,Z_3])=[F_0(Z),\ldots,F_5(Z)]$ where $Q, F_0, \ldots, F_5$ is a basis of
this 7 dimensional vector space.  Show that the morphism $\phi$ is not expressible
by a sextuple of homogeneous polynomials of
the same degree with no common zero on Q.}

\itemitem{Proof:}{(K. Dalili)}

\bye











