Abstract
Let f:X→P1 be a non-isotrivial family of semi-stable curves of genus g≥1 defined over an algebraically closed field k. Denote by snc the number of the singular fibers whose Jacobians are non-compact. We prove that snc≥5 if k=C and g≥5; we also prove that snc≥4 if char(k)>0 and the relative Jacobian of f is non-smooth.
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