= Constancy of line bundle degree in a family
{title2=$t\mapsto\deg(M|_{C\times\{t\}})$}
For a <line bundle> $M$ on $C\times T$, with $C$ a smooth projective curve, this function is locally constant. Twist both $M$ and $M^{-1}$ by a large power of an <ample line bundle> on $C$. Near a chosen fiber their first <sheaf cohomology> groups vanish. The <Riemann-Roch theorem> and upper semicontinuity of their zeroth <sheaf cohomology> groups then bound the degree from above and below by the degree on that fiber. Connectedness of $T$ is necessary for a single constant across all fibers.
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