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

\bigskip


\centerline{ Problem Set 7 (Last revised \number\month/\number\day/\number\year)}
\bigskip

\item{13.1}{Let $X \subset \bP^n$ be a finite set of $d$ points.
Show that for sufficiently large values of $m$ (say $m \geq d-1$) the Hilbert
function $h_X(m)=d$.}

\itemitem{Proof:}{The Hilbert function depends only on the
homogeneous coordinate ring, which is unchanged by projective
transformations, so we may assume all points are in the open set
$z_0\not = 0$ so the points are of the form $p_i=[1,a_{i1},\ldots,a_{in}]$.
Consider the map of vector spaces from homogeneous polynomials of
degree m to $K^d$ given  by $F(z_0,\ldots,z_n)\to
(F(p_1),\ldots,F(p_n))$, where the polynomial is evaluated on the
chosen representatives
above for $p_i$.  By choosing a linear polynomial $L_i$ which is
zero on $p_i$ and nonzero on all other $p_j$ and forming the degree
$d-1$ polynomials $F_j=\prod_{i \not = j}L_i$ we see that by
multiplying these by powers of $z_0$ that the vector space map above
is onto, and kernel is the degree m polynomials in the ideal of
functions vanishing
on $X$. Thus $h_X(m)=d$ for $m \geq d-1$.}



\medskip


\item{13.5}{Find the Hilbert function of a hypersurface of degree d
in $\bP^n$ and verify that the dimension of the hypersurface is $n-1$.}

\itemitem{Proof:}{Let $X=Z(F)$ be a hypersurface, where $F$ is a
square free nonconstant polynomial of degree $d$, so that $I(X)=(F)$. The vector
space of polynomials in $I(X)$ of degree $m$ is zero when $m < d$, and
is
isomorphic to the space of all polynomials of degree $m-d$ (via
multiplication by $F$) when $m \geq d$. Thus for $m \geq d$ we have
$h_X(m)={m+n \choose n} - {m-d+n \choose n}=dm^{n-1}/(n-1)! + \cdots$.
This agrees with the dimension being $n-1$.

\medskip


\item{13.6}{Find the Hilbert function of the Segre variety $\Sigma_{n,m}=
\sigma(\bP^n\times \bP^m) \subset \bP^{nm+n+m}$ and verify that the dimension
is $n+m$.}

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


\medskip

\item{13.8}{Determine the arithmetic genus of (i) a pair of skew lines in $\bP^3$
(i) a pair of incident lines in either $\bP^2$ or $\bP^3$ (iii) three concurrent
but not coplanar lines in $\bP^3$ and (iv) three concurrent coplanar lines 
in either $\bP^2$ or $=bP^3$.}


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

\medskip

\item{13.9}{Consider a plane curve $X \subset \bP^2$ of degree d and its
image $Y=\nu_2(X) \subset \bP^5$ under the quadratic Veronese map.  Compare the
Hilbert polynomials of the two and observe in particular that the arithmetic genus is
the same.}


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

\medskip

\item{13.13}{Verify the possible free resolutions of a configuration of
four points in the plane detailed in example 13.12}

\itemitem{Proof:}{()}

\medskip

\item{13.14}{Find the minimal free resolutions of $\Gamma$ in $\bP^3$ in
case (i) $\Gamma$ consists of the four coordinate points (ii) $\Gamma$
consists of five points, no four coplanar.}

\itemitem{Proof:}{(
%o31 = total: 1 6 8 3
%          0: 1 . . .
%          1: . 6 8 3
% 0\to S(-4)^3 \to S(-3)^8 \to S(-2)^6 \to S \to S(\Gamma)
%                 x_0x_1 x_0x_2 x_1x_2 x_0x_3 x_1x_3 x_2x_3 
%
%

)}
\medskip
\item{13.15}{Find the minimal freee resolution of the twisted cubic curve.}

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

\item{13.17}{Prove a weak form of the Bezout theorem in $\bP^2$: if $F,G$ are
polynomials of degree $d,e$ on $\bP^2$ without common factors such that $F,G$
generate the ideal of their intersection, then the intersection consists of
$d\cdot e$ points.  Similarly show that if $\Gamma \subset \bP^3$ is a complete
intersection of surfaces of degrees $d,e,f$ then $\Gamma$ consists of $d\cdot e\cdot f$
points.}

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


\medskip

\item{13.18}{Find the Hilbert polynomial of a complete intersection in
$\bP^3$ of surfaces of degrees $d,e$.  In case this intersection is a
curve, what is the arithmetic genus of this curve?}

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

\medskip

\item{14.1}{Let $K \simeq \bC$ be the complex field. Prove that $X$ is smooth at $p$ if and only if
$X$ is a smooth complex manifold at $p$ if and only if $X$ is a real
$C^\infty$ submanifold of $\bC^n$ at $p$.}

\itemitem{Proof:}{()}
\medskip

\item{14.2}{Show that the dimension of the Zariski tangent space $T_pX$ is an upper-semicontinuous
function of p in the Zariski topology.}

\itemitem{Proof:}{The dimension of the Zariski tangent space at $p$ to $X \subset \bA^n$
is 
just the n minus the rank of the $l \times n$ Jacobian matrix $\partial f_i /\partial x_j$ 
if $I(X)=(f_1,\ldots,f_l)$.
The dimension of the tangent space is greater than
or equal to a positive integer $m$ if and only if the Jacobian matrix has
rank less than or equal to $n-m$ if and only if all $(n-m+1)\times(n-m+1)$
minors of the Jacobian have determinant zero. This is clearly a set of equations
describing a closed subset of $X$, proving the upper semicontinuity.}

\medskip

\item{14.3}{(i) Show that if $X$ is a hypersurface then the locus of singular
points is a proper subvariety of $X$. (ii) Use this together with the
fact that any variety is birational to a hypersurface (11.23) to prove that
the singular points of any variety form a proper subvariety.}

\itemitem{Proof:}{(i) It is enough to do this for an irreducible hypersurfaces
which is the zero set of a nonconstant  irreducible polynomial $F(x_1,\ldots,x_n)$.
The locus of singular points $X_{sing}$ is the closed set of points 
$(x_1,\ldots,x_n) \in X$
such that $\partial F /\partial x_i$ simultaneously vanish.  If $X_{sing}=X$
then the ideals $(F), (F, \partial F/\partial x_1,\ldots,\partial F/\partial x_n)$
define the same variety so that $\partial F/\partial x_i$ is a multiple of
$F$. The total degree of $\partial F/\partial x_i$ is less than that of $F$,
so it must be true that $\partial F/\partial x_i=0$ as a polynomial.
In characteristic zero it is easy to see that the partial derivative of a
monomial is nonzero if it is not a constant, so this is impossible in
characteristic 0.  In characteristic $p$ the only monomials with all
partial derivatives zero are $p$th powers, and since a $p$th power of
a sum in characteristic $p$ is a sum of $p$th powers we conclude that $F$ is
a $p$th power contradicting that it is irreducible. So in each case $X_{sing}$
is a proper subvariety of $X$, so that its complement $X_{sm}$,
the set of smooth points, is a 
nonempty open subset (which is dense if X is irreducible).
(ii) Since any variety is birational to a hypersurface, it has a a nonempty open subset isomorpic
to an open subset in a hypersurface. By (i), not every point in a nonempty
 open subset in a hypersurface is singular. Hence in an arbitrary variety ther
are always smooth points, and the set of singular points is a proper subvariety
of strictly smaller dimension.}
\medskip

\item{14.5}{Let $X\subset \bA^2$ be the curve $x-y^p=0$ and let $Y=\bA^1$
and let $f:X\to Y$ be given by $f(x,y)=y$. Show that if $K$ has characteristic
$p$ then $X,Y$ are smooth, $f$ is surjective, but $df_q$ is zero for all
$q\in X$.}

\itemitem{Proof:}{()
}
\medskip


\item{14.6}{Let $f:X\to Y$ be any regular map of affine varieties. Show that
the dimension of the kernel of the differential $df_p$ is an upper semicontinous
function of $p$.}


\itemitem{Proof:}{()
}

\medskip

\item{14.7}{Let $X \subset \bA^4$ be the hypersurface $ax^2+y^2=b$, and let $Y=\bA^2$ with coordinates $a,b$ and let $f$ be the 
projection. Show that $X$ is smooth of dimension 3, and that every fiber of $f$ has dimension 1. Find the set of points $p\in X$ such that
$df_p$ has rank $<2$ and verify 14.6 for this map.  At the same time, find the locus of
$p\in X$ such that the Zariski tangent space to $X_{f(p)}$ at $p$ has dimension greater that
1 and show that it is not closed.}


\itemitem{Proof:}{()
}

\medskip



\item{14.11}{(i) Verify directly that the map $h$ does not induce an
isomorphism between the tangent space to $Y$ and to $Z$ at the origin.
(ii) Show that $Y$ and $Z$ are not isomorphic.}


\itemitem{Proof:}{()
}


\medskip


\item{14.12}{Verify that the row space of this matrix is the projective
tangent space to $X$ at $p$.}

\medskip


\item{14.13}{Use Corollary 14.10 to show that the Veronese and Segre
varieties are smooth and verify this directly by looking at their defining
equations. (Alternately, deduce this from 14.3 and the transitive 
action of an algebraic group on these varieties).}


\itemitem{Proof:}{()
}


\medskip


\item{14.14}{Show that the rational normal scroll $X_{a_1,\ldots,a_n} \subset
\bP^n$  is smooth if and only if all $a_i >0$.}

\medskip

\item{14.15}{Let $X\subset \bP^n$ be any ireducible nondegenerate variety (that is $X$ is not contained in any
hyperplane in $\bP^n$) and let $S(X)$ be its secant variety.
Show that the projective tangent space to $S(X)$ at a point of $X$ is all of $\bP^n$.}

\medskip


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