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
    \[ p_g\geq q-12\]
    .