Assuming the Minimal Model Program in dimension n-1, every uniruled but not rationally connected projective log canonical pair of dimension n with pseudoeffective log canonical divisor has a good model.
On Nonvanishing for uniruled log canonical pairs
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abstract
We prove the Nonvanishing conjecture for uniruled projective log canonical pairs of dimension $n$, assuming the Nonvanishing conjecture for smooth projective varieties in dimension $n-1$. We also show that the existence of good minimal models for non-uniruled projective klt pairs in dimension $n$ implies the existence of good minimal models for projective log canonical pairs in dimension $n$.
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2019 1verdicts
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Abundance for uniruled pairs which are not rationally connected
Assuming the Minimal Model Program in dimension n-1, every uniruled but not rationally connected projective log canonical pair of dimension n with pseudoeffective log canonical divisor has a good model.