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Abstract
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In joint work with Gunter Fuchs we show that if L[E] is tame, has no strong cardinal, and does not know how to fully iterate itself, then L[E] has class many grounds and their intersection is a lower part model, the "minimal core" of L[E]. In sharp contrast, in joint work with Grigor Sargsyan, building upon earlier work of himself and Martin Zeman, we show that if L[E] is least with a strong cardinal above a Woodin cardinal, then L[E] has only set many grounds. Its smallest ground is of the form L[E',\Sigma], where L[E'] is the fully iterable (in L[E]) core model of L[E] and \Sigma is an iteration strategy for L[E']. In particular, L[E',\Sigma] has no proper ground.
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