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We give several remarks on a conjecture of Batyrev and Manin on the distribution of rational points on algebraic varieties. We study the signature of the geometric invariant of Batyrev-Manin in relation to the value of the Kodaira dimension, and study the validity of the Batyrev-Manin conjecture under the assumption that there exists an unramified covering for which the Batyrev-Manin conjecture holds.

Remarks on a conjecture of Batyrev and Manin