We study stable curves of arithmetic genus 2 which admit two morphisms of finite degree d, resp. (d,d′), onto smooth elliptic curves, with particular attention to the case d prime.

We construct a smooth algebraic stack of tuples consisting of genus two nodal curves, line bundles, and twisted fields. It leads to a desingularization of the moduli of genus two stable maps to projective spaces. The construction is based on systematical application of the theory of stacks with twisted fields (STF), which has its prototype appeared in arXiv:1906.10527 and arXiv:1201.2427 and is fully developed in this article. The results of this article are the second step of a series of works toward the resolutions of the moduli of stable maps of higher genera.

Let (X, ) be a pair. We study how the condition (X, ) causes surjectivity or birationality of the Albanese map and the Albanese morphism of (X, ) in both characteristic (X, ) and characteristic (X, ) . In particular in characteristic (X, ) we generalize Kawamata's result to the cases of log canonial pairs, and in characteristic (X, ) we generalize a result of Hacon-Patakfalvi to the cases of log pairs. Moreover we show that if (X, ) is a normal projective threefold in characteristic (X, ) , the coefficients of the components of (X, ) are (X, ) and (X, ) is semiample, then the Albanese morphism of (X, ) is surjective under reasonable assumptions on (X, ) and the singularities of the general fibers of the Albanese morphism. This is a positive characteristic analog in dimension 3 of a result of Zhang on a conjecture of Demailly-Peternell-Schneider.

The DK Flip Conjecture of Bondal-Orlov and Kawamata states that there should be an embedding of derived categories for any flip, which is known to be true for toroidal flips. In this paper, we construct new examples of Grassmannian flips that satisfy the DK Flip Conjecture.

Let (X, g) be a compact Riemannian manifold with quasi-positive Riemannian scalar curvature. If there exists a complex structure J compatible with g, then the Kodaira dimension of (X, J) is equal to −∞ and the canonical bundle KX is not pseudo-effective. We also introduce the complex Yamabe number λc(X) for compact complex manifold X, and show that if λ_c(X) is greater than 0, then κ(X) is equal to −∞; moreover, if X is also spin, then the Hirzebruch A-hat genus \hat{A}(X) is zero.