Applications of tight closure and F-singularity to algebraic geometry

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Given a pair of a variety of characteristic p and an effective divisor on it, one can associate a real number called the F-pure threshold. Since this invariant is defined as a characteristic p analog of the log canonical threshold in characteristic 0, it is desirable that F-pure thresholds are rational numbers similarly as log canonical thresholds. N.Hara studied F-pure thresholds of pairs of a nonsingular surface and an effective divisor, and proved based on Monsky's idea of p-fractals that the F-pure thresholds are rational provided that the base field is finite. When the divisor is defined by a homogeneous polynomial f (x, y), the F-pure threshold c(f) can be estimated more precisely, and we can obtain a finite list of possible value of c(f) for a fixed degree d=deg f and characteristic p. We also proved that the Monsky's function ψ_f(t) has a piecewise quadratic limit as p→∞.M.Ishida studied real fans from a viewpoint of toric geometry, as well as moduli parameter of Catanese-Ciliberto-Ishida surface. T.Kajiwara studied the theory of logarithmic abelian varieties, the relationship of tropical hypersurfaces and degeneration of projective toric varieties, and the theory of tropical toric varieties. K.-i.Watanabe studied geometric interpretation of integrally closed monomial ideals in 3 variables, multiplier ideals, and F-thresholds. K.Yoshida gave estimates of multiplicities of Stanley-Reisner rings and Buchsbaum homogeneous algebras, and studied the structure of these rings when they have minimal multiplicities.