Campana introduced a notion of Campana rational connectedness for Campana orbifolds. Given a Campana fibration over a complex curve, we prove that a version of weak approximation for Campana sections holds at places of good reduction when the general fiber satisfies a slightly stronger version of Campana rational connectedness. Campana also conjectured that any Fano orbifold is Campana rationally connected; we verify a stronger statement for toric Campana orbifolds. A key tool in our study is log geometry and moduli stacks of stable log maps.
Let A = (a_1,…,a_n)∈ℤ^n be a sequence with sum k(2g-2+n). The double ramification cycle 𝖣𝖱_g(A) ∈𝖢𝖧^g(ℳ̅_g,n) is the virtual class of the locus of curves (C,p_1,…,p_n) where the line bundle (ω_C^log)^-k(∑ a_i p_i) is trivial. Although there has long been a formula for 𝖣𝖱_g(A) [JPPZ17], the exact dependence on A was unknown for a long time, though it was conjectured to be polynomial in A. A proof was announced in [JPPZ17], and Pixton gave a proof incorporating ideas of Zagier in [Pix23]. Here we present an alternative proof of the polynomiality of the double ramification cycle.
Hilbert polynomials have positivity properties under favorable conditions. We establish a similar "K-theoretic positivity" for matroids. As an application, for a multiplicity-free subvariety of a product of projective spaces such that the projection onto one of the factors has birational image, we show that a transformation of its K-polynomial is Lorentzian. This partially answers a conjecture of Castillo, Cid-Ruiz, Mohammadi, and Montano. As another application, we show that the h*-vector of a simplicially positive divisor on a matroid is a Macaulay vector, affirmatively answering a question of Speyer for a new infinite family of matroids.
