We propose an explicit description of the exceptional set in Manin's Conjecture. Our proposal includes the rational point contributions from any subvariety or cover with larger geometric invariants. We prove that this set is contained in a thin subset of rational points, verifying there is no counterexample to Manin's Conjecture which arises from incompatibility of geometric invariants.the setis contained in a thin subset of X(F ).
We conjecture that the exceptional set in Manin's conjecture has an explicit geometric description. Our proposal includes the rational point contributions from any generically finite map with larger geometric invariants. We prove that this set is contained in a thin subset of rational points, verifying that there is no counterexample to Manin's conjecture which arises from an incompatibility of geometric invariants.
We show that the b-constant (appearing in Manin's conjecture) is constant on very general fibers of a family of algebraic varieties. If the fibers of the family are uniruled, then we show that the b-constant is constant on general fibers.
In this paper, we provide counterexamples to Mercat's conjecture on vector bundles on algebraic curves. For any n ≥ 4, we provide examples of curves lying on K3 surfaces and vector bundles on those curves which invalidate Mercat's conjecture in rank n.
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