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In this work, we show that for a certain class of threefolds in positive characteristics, rational-chain-connectivity is equivalent to supersingularity. The same result is known for K3 surfaces with elliptic fibrations. And there are examples of threefolds that are both supersingular and rationally chain connected.

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An effective algorithm of determining Gromov–Witten invariants of smooth hypersurfaces in any genus (subject to a degree bound (1)) from Gromov–Witten invariants of the ambient space is proposed.

The main goal of this paper is to introduce a set of conjectures on the relations in the tautological rings. In particular, the framework gives an efficient algorithm to calculate all tautological equations using only finite dimensional linear algebra. Other applications are also indicated.