  In the first part of this article, we answer a question of I. Dolgachev,
which is related to the following problem: given a birational map
$f\in\mathrm{Bir}(\mathbb{P}^m_\mathbf{k})$ and a linear projective map
$A\in\mathrm{PGL}_{m+1}(\mathbf{k})$, when is $A\circ f$ regularizable?
Dolgachev's initial question is whether this may happen for all $A$ in
$\mathrm{PGL}_{m+1}(\mathbf{k})$, and the answer is negative. We then look at
the sequence $n\mapsto\mathrm{deg}\, f^n$,
$f\in\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$. We show that there is no
constraint on the sequence $n\mapsto\mathrm{deg}\, f^n-\mathrm{deg}\, f^{n-1}$
for small values of $n$. Finally we study the degree of pencils of curves which
are invariant by a birational map. When f is a Halphen or Jonqui\`eres twist,
we prove that this degree is bounded by a function of $\mathrm{deg}\, f$. We
derive corollaries on the structure of conjugacy classes, and their properties
with respect to the Zariski topology of
$\mathrm{Bir}(\mathbb{P}^2_\mathbf{k})$.
