A criterion of exactness of the Clemens-Schmid sequences arising from semi-stable families of open curves (Q1423473)

Let \(\pi :X\rightarrow \Delta\) be a semistable family of projective algebraic curves over the unit disk (\(\pi\) is flat and projective over \(\Delta\), smooth over \(\Delta ^*=\Delta \backslash \{ 0\}\), and \(Y=\pi ^{-1}(0)\) is a divisor with normal crossings and without multiple components). Set \(X_t=\pi ^{-1}(t)\) (\(t\in \Delta ^*\)), \(X^*=X\backslash Y\). The Clemens-Schmid sequence \[ \cdots \rightarrow H^{q-2}(X_t,{\mathbb Q})\rightarrow H_Y^q(X,{\mathbb Q}) \rightarrow H^q(Y,{\mathbb Q})\rightarrow H^q(X_t,{\mathbb Q}) @>{N}>> H^q(X_t,{\mathbb Q})\rightarrow \cdots \] where \(N\) is the log of the monodromy action, is exact. The author gives necessary and sufficient conditions the sequence to be exact in the case when \(\pi\) is a semistable family of open curves. He shows that in general the sequence is not exact for non-proper families.