Algebraic Geometry and Linear Algebra

Gian Pietro Pirola
University of Pavia (Italy)
Exercise 1. Let 
 be a field and 
 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 
Consider the cases
\[K=\mathbb{C}, \mathbb{R}\]
  1. Show that
    is closed in
    both with the standard and with the Zariski topology.
  2. Take
    . Show that
    is irreducible (Zariski topology). Is
    connected in the standard topology?
  3. Set
    . Show that any vector subspace of dimension 2 of
    \[M_{n, n}(K)\]
    \[D_{n-1}\setminus \{0\}\]
    . Show that this fails for
    if and only if
    is even, and for
    for any
  4. Take
    . Show that
    is a complex variety of dimension
Exercise 2. Let
be a complex vector space of dimension
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: 
  1. Show that
    has not non trivial holomorphic sections (sections of
  2. Does the dual
    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
Exercise 3. Let
be a vector bundle of rank 2 having Chern polynomial
. 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 
    of rank 
  2. \[Sym^m(E)\subset E^{\otimes m}\]
Exercise 4. Let
be a projective surface. Set
\[H^{p, q}(S)=H^q(S, \Omega^p)\]
, where
is the holomorphic cotangent bundle of
, and consider the maps
\[\phi:\wedge^2H^{1,0}(S)\to H^{2,0}(S)\]
\[\mu:H^{1, 0}(S)\otimes H^{0, 1}(S)\to H^{1, 1}(S)\]
. Let
be the kernel of
be the kernel of
. 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
    does not contain decomposable elements. Show that
    . Show that also
    does not contain decomposable elements and that
    \[h^{11}\geq 2q\]
  2. Assume that
    does not contain rank 4 elements. Show that
    \[ p_g\geq q-12\]