  Let $F$ be a finite unramified extension of $\mathbb Q\_p$ and $\bar\rho$ be
an absolutely irreducible mod~$p$ $2$-dimensional representation of the
absolute Galois group of $F$. Let $t$ be a tame inertial type of $F$. We
conjecture that the deformation space parametrizing the potentially
Barsotti--Tate liftings of $\bar\rho$ having type $t$ depends only on the Kisin
variety attached to the situation, enriched with its canonical embedding into
$(\mathbb P^1)^f$ and its shape stratification. We give evidences towards this
conjecture by proving that the Kisin variety determines the cardinality of the
set of common Serre weights $D(t,\bar\rho) = D(t) \cap D(\bar\rho)$. Besides,
we prove that this dependance is nondecreasing (the smaller is the Kisin
variety, the smaller is the number of common Serre weights) and compatible with
products (if the Kisin variety splits as a product, so does the number of
weights).
