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{Mathematics 535 \hfill Algebraic Geometry \hfill Fall 2008}

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\centerline{ Problem Set 7B (Last revised \number\month/\number\day/\number\year)}
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\item{18.2}{Assuming that the characteristic of $K$ is 0, use exercises 7.16 and 11.44 to
show that part (iii) of Definition 18.1 is well-defined, and use this
to show that parts (i) and (ii) are also well-defined.}

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\item{18.7}{Let $X \subset \bP^n$ be any subvariety of pure dimension
of degree 1.  Show that $X$ is a linear subspace of $\bP^n$.}
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\item{18.11}{Give a proof of the nondegeneracy of the general hyperplane
section of the variety $X$ without invoking the notion of degree or
Bezout's theorem as follows. First show that if the general hyperplane
section of $X$ spans a $k$-plane we have a rational map
$\phi:\bP^{n*}--\to \bG(k,n)$ defined by sending a general hyperplane
$H\in\bP^{n*}$ to the span of $H\cap X$.  Next use the fact that the
universal hyperplane section of $X$ is irreducible to deduce that for
any $H\in\bP^{n*} $ and any point $\Lambda \in \phi(H)$ the hyperplane
section $H\cap X$ lies on the $k$-plane $\Lambda$.  It follows that if
the general hyperplane section is degenerate then all are. But any $n$
independent points of $X$ will span a hyperplane $H$ with $H \cap X$
nondegenerate.}

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\item{18.14}{Let $Y\subset \bP^n$ be a variety of dimension $k$
and degree e, and $Z=\nu_d(Y)$ its image in $\bP^N$ under the Veronese
map.  Use the various degree computing techniques to compute the
degree of $Z$.  }


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\item{18.18}{Let K have characteristic 0. Let $X,Y\subset \bP^n$ be disjoint
subvarieties of dimensions $k$ and $l$, and $\Gamma \subset \bP^n$ a general
plane of dimension $n-k-l-1$. Show that the projections $\pi_\Lambda(X)$
and $\pi_\Gamma(X)$ intersect transversely.}



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