On the Space of Conics on Complete Intersections

Abstract

We get sharp degree bound for generic smoothness and connectedness of the space of lines and conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on hypersurfaces. As a consequence, we prove that for a Fano complete intersection $$X$$ with index $$\ge 2$$ , the $$1$$ -Griffiths group generated by algebraic $$1$$ -cycles homologous to $$0$$ modulo algebraic equivalence is trivial, which is a conjecture for general rationally connected varieties.