Monday, 30 May 2011

ag.algebraic geometry - Extending maps of curves

(I'm happy to work over an algebraically closed field....)



Let mathcalCrightarrowSpec(R) be a (flat) family of proper, prestable curves where R is a DVR. Suppose the generic fiber is smooth and the special fiber, C0, is reduced but may be reducible.



Given a finite map of curves f0:D0rightarrowC0 with D0 also prestable, can this be extended to some map on some family?



That is, is there a flat family of proper curves mathcalDrightarrowSpec(R) and an R-morphism f:mathcalDrightarrowmathcalC which reduces to f0 on the special fiber?



Perhaps such an extension is possible only after a ramified cover of Spec(R)?



If so, can it be arranged that the generic fiber of mathcalD is smooth?

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