Algebraic Geometry and Linear Algebra

Gian Pietro Pirola
University of Pavia (Italy)

Exercises

Exercise 1. Let
$K$
be a field and
$M=M_{n,m}(K)$
be the vector space of
$n\times m$
matrices. Consider the set
$D_k(K)=D_k=\{A\in M:\ {\rm rank}(A)\leq k\}$
and let
$U_k(K)=U_k=D_k\setminus D_{k-1}$
be the subset of rank
$k$
matrices.
Consider the cases
$K=\mathbb{C}, \mathbb{R}$
and
$\mathbb{Q}$
respectively.
1. Show that
$D_k$
is closed in
$M_{n,m}(K)$
both with the standard and with the Zariski topology.
2. Take
$n=m=3$
. Show that
$D_2(K)$
is irreducible (Zariski topology). Is
$D_2(K)$
connected in the standard topology?
3. Set
$n=m>1$
and
$K=\mathbb{C}$
. Show that any vector subspace of dimension 2 of
$M_{n, n}(K)$
intersects
$D_{n-1}\setminus \{0\}$
. Show that this fails for
$K=\mathbb{R}$
if and only if
$n$
is even, and for
$K=\mathbb{Q}$
for any
$n$
.
4. Take
$K=\mathbb{C}$
. Show that
$U_k(\mathbb{C})$
is a complex variety of dimension
$k(n+m-k)$
.
Exercise 2. Let
$\mathbb{C}^{N+1}$
be a complex vector space of dimension
$N+1$
and
$\mathbb{P}^N$
be the associated projective space of lines. Consider
$S\subset \mathbb{C}^{N+1}\times \mathbb{P}^N$
defined by
$S=\{(v, L):\ v\in L\}.$
The projection
$\pi:S\to \mathbb{P}^N$
defines a line bundle:
$\mathcal{O}(-1).$
1. Show that
$\mathcal{O}(-1)$
has not non trivial holomorphic sections (sections of
$\pi$
).
2. Does the dual
$\mathcal{O}(1)$
have non zero holomorphic sections?
3. Consider the inclusion
$\mathcal{O}(-1)\subset \mathbb{C}^{N+1}\times \mathbb{P}^N=\mathcal{O}^{N+1}$
and compute of the Chern classes of the quotient
$\mathcal{O}^{N+1}/\mathbb{O}(-1)$
.
Exercise 3. Let
$E$
be a vector bundle of rank 2 having Chern polynomial
$c(E)=1+tc_1(E)+t^2c_2(E)$
. Compute the Chern classes of
$E\otimes E$
and of
$Sym^2(E)\subset E\otimes E$
the symmetric vector bundle (Hint: use the splitting principle, that is assume
$E=A\oplus B$
, and the formula
$c(F\oplus G)=c(F)c(G)$
). Generalize to the cases
1.
$E$
of rank
$n,$
2. $Sym^m(E)\subset E^{\otimes m}$
.
Exercise 4. Let
$S$
be a projective surface. Set
$H^{p, q}(S)=H^q(S, \Omega^p)$
, where
$\Omega^1$
is the holomorphic cotangent bundle of
$S$
and
$\Omega^p=\wedge^p\Omega^1$
, and consider the maps
$\phi:\wedge^2H^{1,0}(S)\to H^{2,0}(S)$
and
$\mu:H^{1, 0}(S)\otimes H^{0, 1}(S)\to H^{1, 1}(S)$
. Let
$K$
be the kernel of
$\phi$
and
$C$
be the kernel of
$\mu$
. Let
$q={\rm dim}H^{1, 0}(S),\ p_g={\rm dim}H^{2, 0}(S),\ h^{11}={\rm dim}H^{1, 1}(S)$

1. Assume that
$K$
does not contain decomposable elements. Show that
$p_g>2q-3$
. Show that also
$C$
does not contain decomposable elements and that
$h^{11}\geq 2q$
2. Assume that
$K$
does not contain rank 4 elements. Show that
$p_g\geq q-12$
.