Direct images of vector bundles on rational surfaces (Q1972284)

The paper studies a new method to investigate the cohomology of vector bundles \({\mathcal E}\) over blow-ups \(X_r\) of \({\mathbb P}^2\) at a set of \(r\) points. The main idea is to consider the pencil of lines through one of the points which are the base points of the blow-up; this induces a map \(\pi : X_r \rightarrow {\mathbb P}^1\) whose fibers are the direct images of the lines (generically, there will be \(r-1\) singular fibers). If \({\mathcal E}\) satisfies mild hypotheses, the author shows that all the dimensions of its cohomology modules are computable on \({\mathbb P}^1\), namely: \(H^*(X_r,{\mathcal E}) \cong H^*({\mathbb P}^1,\pi _*{\mathcal E})\). Of course the main advantage is that on \({\mathbb P}^1\) every bundles splits, i.e. \(\pi _* {\mathcal E} \cong \bigoplus _{k=1}^R{\mathcal O}_{{\mathbb P}^1}(a_k)\), hence the problem of computing the cohomology of \({\mathcal E}\) reduces to that of determining the \(a_k\)'s. In particular, this technique could be of use in attacking a well known problem, namely to determine the dimension of the linear system of plane curves of degree \(d\) which have multiplicity at least \(m_i\) at the blown-up point \(P_i\) (for generic \(P_i\)'s); this is equivalent to determine \(H^0(X_r,{\mathcal L}_{d,{\mathbf m}})\), where \({\mathbf m}=(m_1,\dots{},m_r)\in {\mathbb Z}^r\) and \({\mathcal L}_{d,{\mathbf m}}\cong {\mathcal O}_{X_r}(C)\) is the line bundle associated to the strict transform \(C\) of such a curve in \({\mathbb P}^2\). In the paper, bounds are given for the \(a_k\)'s for small \(r\) or small \(m_i\)'s and the long standing conjecture about the problem above (by B. Segre, B. Harbourne, A. Hirschowitz and the reviewer) is reformulated in term of the \(a_k\)'s.