RANK 2 VECTOR BUNDLES IN A
NEIGHBORHOOD OF AN EXCEPTIONAL
CURVE OF A SMOOTH SURFACE

E. BALLICO

Abstract:

Let $D\cong{\bf P}^1$ be an exceptional divisor on the smooth surface W and U the formal neighborhood of D in W. Let E be a rank 2 vector bundle on U. Here we associate to E an integer $t\ge1$, a finite family E_i, $1\le i\le t$, of rank 2 vector bundles on U and a finite sequence $\{(a_i,b_i)\}_{1\le i\le t}$ of pairs of integers such that E_i|D has splitting type (a_i,b_i), E₁=E, a_t=b_t, a_i+1+b_i+1=a₁+b₁+i and $b_i<b_{i+1}\le a_{i+1}\le a_i$ for $2\le i\le t$. Vice versa, for any such sequence we prove the existence of at least one such bundle. We compute the second Chern class of E in terms of $\{(a_i,b_i)\}_{1\le i\le t}$ and show that ${\bf O}_U(-a_1D)\oplus {\bf O}_U(-b_1D)$ is the unique bundle with splitting type (a₁,b₁) and maximal c₂.