\magnification=\magstep1
\def \bA {{\bf A}}
\def \bP {{\bf P}}
\def \bZ {{\bf Z}}
\def \bG {{\bf G}}
\def \bQ {{\bf Q}}
\def \bO {{\bf O}}
\def \bF {{\bf F}}
\def \bT {{\bf T}}
\def \bP {{\bf P}}


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

\bigskip


\centerline{ Problem Set 4 (Last revised \number\month/\number\day/\number\year)}
\bigskip

\item{5.2}{Show that he following are equivalent for homogeneous
ideals $I, J \subset K[Z_0,\ldots , Z_n]$
$$\cases {(i)& $ I$  and $J$  have the same saturation\cr
        (ii)& $I_m=J_m$ for all $m$ sufficiently large\cr
        (iii)& $I,J$ generate the same ideal in each localization $K[Z_0,\ldots,Z_n,
Z_i^{-1}]$\cr}$$
}
\itemitem{Proof:}{((K. Dalili)  )}

\medskip

\item{5.3}{Show that the homogeneous quadratic polynomials
$$F_{i,i}(Z)=Z_i^2-Z_{i-1}Z_{i+1}$$
and $$F_{i,i+1}(Z)=Z_iZ_{i+1}-Z_{i-1}Z_{i+2}$$
for $i=1,\ldots,d-1$ generate the ideal of the rational normal curve locally
but do not generate the homogeneous ideal $I(C)$.}
\itemitem{Proof:}{(J. Burdges)}

\medskip

\item{5.4}{Show that the polynomials $F_{i,j}(Z)=Z_iZ_j-Z_{i-1}Z_{j+1}$
for $1 \leq i \leq j\leq d-1$ generate the homogeneous ideal of the rational
normal curve.  Show that the equations we saw earlier for the Veronese
and Segre varieties generate their homogeneous ideals.}
\itemitem{Proof:}{(T. Pham)}

\medskip

\item{5.5}{Show that a variety $X$ is irreducible if and only if every 
nonempty Zariski open subset is dense if and only if every pair
of nonempty Zariski open subsets meet.}
\itemitem{Proof:}{Two open subsets are disjoint if and only if the union
of their complements is the whole space, since $X - (A \cap B) =
(X-A)\cup (X-B)$. Hence irreducible is equivalent to every two nonempty
open sets meet.  In any topological space this is equivalent to the property
that every nonempty open set has closure equal the whole space.} 


\medskip

\item{5.9}{Show that the product of two irreducible varieties is irreducible.
Is the intersection of irreducible varieties irreducible?}
\itemitem{Proof:}{Let $X\subset \bP^m, Y\subset \bP^n$ be irreducible 
varieties.  Suppose that
$X\times Y =\cup \Psi_i$ is the expression of the product as a union of
irreducibles. Let $\Phi_i=\{p \in X | (p,Y) \subset \Psi_i\}$.
Since the variety $\{p\}\times Y$ is isomorphic to $Y$ and is irreducible,
the union of $\Phi_i$ is X (an irreducible subset of a topological space
is contained completely in some irreducible component). The sets $\Phi_i$
are clearly closed subsets of X, being the simultaneous zeros in the variables
in X of the
polynomials describing $\Psi_i$ as the Y variables are specialized to all the
points of Y.  Hence, since $X$ is irreducible, some $\Phi_{i_0}=X$
which implies that $X\times Y \subset \Psi_{i_0}$ and hence is irreducible.
For quasiprojective varieties we observe that a quasiprojective variety
is irreducible if and only if the closure of the variety in projective
space is irreducible, and the the closure of the product of two
quasiprojective varieties is just the product of their closures.}
 
\itemitem{}{A line $Z_1=0$ and a smooth conic $Z_0Z_2-Z_1^2=0$ are irreducible in $\bP^2$.
They intersect in the set $[1,0,0],[0,0,1]$ which is a reducible variety.}

\medskip

\item{5.10}{Consider the parameter space for lines in $\bP^n$ from Example 4.15.
Show that given any linear space $\Gamma \subset \bP^n$ of dimension $n-2$
or less the general line in $\bP^n$ does not meet $\Gamma$.  Show that
given a twisted cubic $C\subset \bP^3$ the general line in
$\bP^3$ does not meet C.}
\itemitem{Proof:}{(B. Kennedy)}
\medskip

\item{5.13}{Show that for any  $d, n\leq (d+1)(d+2)/2$ a general set of n 
points in $\bP^2$ imposes independent conditions on curves of degree d. This means
that the space of polynomials vanishing at the points has codimension
n in the space of all homogeneous polynomials of degree d.
In case $n \leq 2d+1$ what open set in $(\bP^2)^n$ is implicitly referred
to here.}
\itemitem{Proof:}{(F. Louro)}



\medskip

\item{5.14}{Let $C \subset \bP^n$ be a curve of degree d, that is the zero set of
a homogeneous polynomial $F(Z_0,Z_1,Z_2)$ of degree d, which has
no repeated factors in its unique factorization into irreducibles.  Show
that the general line in $\bP^2$ will intersect C in d points.}
\itemitem{Proof:}{(J. Tedor)}


\medskip


Note: Recall that $G(k,n)$ denotes the Grassmannian of k dimensional
subspaces of an n-dimensional vector space, and that $\bG(k,n)$
denotes the set of k dimensional linear spaces in $\bP^n$ (that is
the set $G(k+1,n+1)$).
\medskip

\item{6.2}{Suppose that the characteristic of K is not 2.
Show that a vector $\omega \in \bigwedge^2 V$ is decomposable if
and only if $\omega \wedge \omega=0$ and hence the Grassmannian G(2,V) of
2 planes in V  is cut out by quadrics in $\bP(\bigwedge^2V)$.}

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

\medskip
\item{6.4}{For any point $p \in \bP^3$ and any plane
$H \subset \bP^3$ containing p let $\Sigma_{p,H} \subset \bG(1,3)$ be
the set of lines in $\bP^3$ passing through p and lying in H.  Show that
under the Plucker imbedding $\bG(1,3) \rightarrow \bP^5$ the
subvariety $\Sigma_{p,h}$ is
mapped to a line, and conversely every line in $\bP^5$ lying on $\bG(1,3)$ is
of the form $\Sigma_{p,H}$ for some p and H.}

\itemitem{Proof:}{ see 6.9
}
\medskip

\item{6.5}{For a point $p \in \bP^3$ let $\Sigma_p \subset \bG(1,3)$
be the set of lines in $\bP^3$ passing through p.  For any plane
$H \subset \bP^3$ let $\sigma_H \subset \bP^3$ be the set of lines
in $\bP^3$ contained in H.  Show that the Plucker embedding carries
both $\Sigma_p, \Sigma_H$ into 2--planes in $\bP^5$.  Show that
conversely any 2--plane $\Lambda\simeq \bP^2 \subset \bG(1,3)
 \subset \bP^5$ either equals $\Sigma_p$ for some p or $\Sigma_H$
for some H. }

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

\item{6.6}{Let $\ell_1,\ell_2 \subset \bP^3$ be skew lines (that is nonintersecting
lines).  Show that the set $Q\subset \bG(1,3)$ of lines in $\bP^3$ meeting
both is the intersection of $\bG(1,3)$ with a three--plane $\bP^3\subset
\bP^5$ and hence is a quadric surface. Deduce that $Q$ is isomorphic to $\bP^1\times\bP^1$.
Do the same problem for lines $\ell_1,\ell_2 \subset \bP^3$ which meet.}

\itemitem{Proof:}{(F. Louro))}

\medskip

\item{6.7}{Let $Q\subset \bP^3$ be a smooth quadric surface.  Show that the
two families of lines on $Q$ correspond to plane conic curves of $\bG(1,3)$
lying in complementary two-planes $\Lambda_1,\Lambda_2 \subset \bP^5$.
Conversely, show that the lines in $\bP^3$
corresponding to a plane conic curve $C \subset \bG(1,3)$
sweep out a smooth quadric surface if and only if the plane $\Lambda$ spanned
by $C$ is not contained in $\bG(1,3)$.  Describe this correspondence if
either the quadric is a cone or the plane $\Lambda$ is in $\bG(1,3)$.}

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

\item{6.8}{let $\Sigma_{1,k} \simeq \bP^1\times \bP^k \subset \bP^{2k+1}$
be the Segre variety, and let $\Lambda_p$ be the fiber over $p \in \bP^1$.
Show that $\Lambda_p$ is a k-plane in $\bP^{2k+1}$ and that the
assignment $p \mapsto \Lambda_p$ gives a regular map of $\bP^1$ to the
Grassmannian $\bG(k,2k+1)$ with image a rational normal curve lying in a
(k+1)-plane in $\bP(\bigwedge^{k+1}K^{2k+2})$.}

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

\medskip



\item{6.9}{Let $G=G(k,V)\subset \bP^N=\bP(\bigwedge^k V)$. (i) Show that
the line $\overline{\Lambda,\Lambda'}$ spanned by any two points
$\Lambda,\Lambda'\in G \subset \bP^N$  lies in G if and only if the corresponding k-planes
intersect in a (k-1)-plane (this is equivalent to having the k-planes lying
in a (k+1)-plane). Hence any line $L \subset G \subset \bP^N$ consists of the set of k-planes in V
containing a fixed (k-1)-plane $\Gamma \subset V$ and contained in a fixed
(k+1)-plane $\Omega \subset V$. (ii) Use the preceding to show that any maximal linear space
$\Phi \subset G \subset \bP^N$ is either the set of k-planes containing a
fixed linear subspace of V or the set of k-planes contained in a fixed linear
subspace of V.}

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

\medskip


\item{6.11}{Let $\Lambda \subset \bP^n$ be a k-plane and let
$I(\Lambda)_d$ be the graded piece of degree d in the homogeneous
ideal of all homogeneous polynomials vanishing on $\Lambda$. Show that
the codimension of $I(\Lambda)_d$ in the vector space $S_d$ of all homogeneous
polynomials of degree d is ${k+d \choose d}$.  Show that the map 
$$\nu_d:G(K,n) \to G({k+d\choose d},{n+d\choose d})$$ obtained by
associating to a k-plane the subspace of linear functionals
on $S_d$ which vanish on $I(\Lambda)_d$ is regular.}

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

\item{6.15}{Let $X \subset \bP^n$ be locally closed. Show that the closure
in $\bG(k,n)$ of the set of k-planes meeting X is the same as the variety of
k-planes meeting the closure of X.}

\itemitem{Proof:}{   Let $\Lambda \subset \bP^n \times
\bG(k,n)$ be the incidence relation $\Lambda=\{(p,V)|p \in V\}$, and let
$\pi_1,\pi_2$ be the projections on $\bP^n, \bG(k,n)$. 
The projective variety $C_k(\bar X)$
of k-planes meeting $\bar X$ is $\pi_2(\pi_1^{-1}(\bar X))$.
Note
$X \subset \pi_1(\pi_2^{-1}(\overline {C_k(X)})) \subset \pi_1(\pi_2^{-1}({C_k(\bar X)})=\bar X $.
Since the image and preimage of regular maps of projective varieties is closed,
the latter three sets are equal. Apply $\pi_2\circ\pi_1^{-1}$ to conclude the result.
}
\medskip

\item{6.16}{(i) Let $C \subset \bP^2 \subset \bP^3$ be the conic
$Z_3=0=Z_0Z_2-Z_1^2$. Find the equation of the variety ${\cal C}_1(C)
\subset \bG(1,3)$ of all incident lines. (ii) Same problem for
the twisted cubic $D=\{[u^3,u^2t,ut^2,t^3]\}$.}
 
\itemitem{Proof:}{(J. Burdges)
}
\medskip


\item{6.18}{Show that if $X, Y\subset \bP^n$ are two varieties contained in complementary linear
subspaces $\Gamma,\Lambda \subset \bP^n$ the join $J(X,Y)$ (the union of
all lines joining X to Y) is 
the intersection of the cones $\overline{\Gamma,Y} \cap \overline{\Lambda,X}$
Next show that any pair of disjoint varieties can be realized 
as the projections of varieties $\tilde X, \tilde Y\subset \bP^n$ 
(contained in complementary subspaces)
projected from $\bP^N$ to $\bP^n$.  Verify that $J(X,Y)$ is
the projection of $J(\tilde X, \tilde Y)$, and hence is a variety.
Can this approach deal with the join of varieties that intersect?}

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

\item{6.20}{Let $Q\subset \bP^3$ be the zero set of $Z_0Z_3-Z_1Z_2$.
Show that the Fano variety $F_1(Q)$ of all lines contained in Q is a union
of two conic curves. Compare this with the parametric description
of the Fano variety given by using the Segre map.}

\itemitem{Proof:}{Consider a line in $\bP^3$ spanned by two vectors
$[a_0,a_1,a_2,a_3],[b_0,b_1,b_2,b_3]$ which is contained in the
quadric Q. The quadric Q is mapped to itself by the group of linear transformations
generated by the operations of switching $Z_0, Z_3$,switching $Z_1, Z_2$,
and simultaneously switching $Z_0,Z_2$ and $Z_1,Z_3$ so it is enough to
find lines on Q such that the matrix 
$$\pmatrix {a_0 &a_1& a_2& a_3\cr b_0&b_1&b_2&b_3}$$
has either its first and fourth or first and third column minors invertible.
In the first case we can take generators for the line to be
$[1,a,b,0], [0,c,d,1]$ and work out the condition for the line to
be on Q. This occurs if and only if three points $[1,a,b,0], [0,c,d,1],
[1,a+c,b+d,1]$ are on Q, that is $ab=0=cd, (a+c)(b+d)=1$.  In terms of
the Plucker coordinates $Z_{ij}$ this is just the equations 
$Z_{24}Z_{34}=0=Z_{12}Z_{13}, 1=(Z_{24}+Z_{12})(Z_{34}+Z_{13})$.  When we
make this homogeneous in the variables we obtain
$Z_{24}Z_{34}=0=Z_{12}Z_{13}, Z_{14}^2=(Z_{24}+Z_{12})(Z_{34}+Z_{13})$.  
So the equations $Z_{24}Z_{34}=0=Z_{12}Z_{13}, 
Z_{14}^2=Z_{24}Z_{13}+Z_{34}Z_{12}$ in the Grassmannian describe lines
on Q. Similarly the other case leads to equations $Z_{34}=0,Z_{12}=0,Z_{23}=Z_{14}$
which together with the Grassmanian equation $Z_{12}Z_{34}-Z_{13}Z_{24}+
Z_{14}Z_{23}=0$ show that we get no new lines
. Thus the Fano variety of lines in Q is the variety in $\bP^5$ given by
the zero set of 
$$Z_{24}Z_{34},Z_{12}Z_{13}, 
Z_{14}^2-Z_{24}Z_{13}-Z_{34}Z_{12},
Z_{12}Z_{34}-Z_{13}Z_{24}+Z_{14}Z_{23}.
$$ This is clearly the intersection of the Grassmanian with the two
planes $Z_{12}=Z_{34}=Z_{14}-Z_{23}=0$ and $Z_{13}=Z_{24}=Z_{14}+Z_{23}=0$,
and in these planes the intersection is respectively the conics
$Z_{13}Z_{24}=Z_{14}^2$ and $Z_{12}Z_{34}=Z_{23}^2$.  So the Fano
variety is the union of two plane conics.  In the parametric description
the two families of lines are those given parametrically for fixed $[a,b]$
by
$[ta,tb,ua,ub$ and $[at,au,bt,bu]$  respectively. The Plucker coordinates
of the first family of lines satisfy $Z_{12}=0=Z_{34},Z_{14}=Z_{23}$
and for the second $Z_{13}=0=Z_{24},Z_{14}=-Z_{23}$, so these two families
are just the intersections of the two planes with the Grassmannian described
above.  See also 6.7.
}

\medskip

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