The rationally connected inequality for the delta-invariant

Let XX be a geometrically rationally connected smooth projective variety of dimension nn, let LL be a big and nef Q\mathbb Q-divisor on XX, let

a(X,L)=inf{tRKX+tLEff1(X)},a(X,L)=\inf\{t\in\mathbb R\mid K_X+tL\in\overline{\operatorname{Eff}}^1(X)\},

and let δ(X,L)\delta(X,L) be the delta-invariant defined via the stable base locus of divisors on the blow-up of the diagonal in X×XX\times X. Rationally connected delta-invariant conjecture. One has

a(X,L)2nδ(X,L).a(X,L)\leq 2n\delta(X,L).

The statement is motivated by Manin's conjecture and would compare the expected exponent of rational-point growth with the paper's general upper bound. It is presented as an expectation, with no resolution supplied in the source.

Sources & referencesView supporting material

Primary source

Sho Tanimoto, “On upper bounds of Manin type”, arXiv:1812.03423 (2019).

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