Stable logarithmic maps to Deligne--Faltings pairs I

Boundedness of the space of stable logarithmic maps

“…The two statements above are proven in this paper by reducing to the case where the characteristic monoid M is globally generated. This case was shown in [AC,Corollary 3.11], by further reducing to the rank one case treated in [Che14].…”

“…A better behaved substack of minimal prestable logarithmic maps is defined in [Che14,AC] when the characteristic sheaf M X is globally generated, in [GS13] when X is a Zariski logarithmic scheme, and in [Wis14] in general. It is denoted M(X).…”

$$\mathbb {A}^1$$-connected varieties of rank one over nonclosed fields

“…When the non-proper variety admits a log smooth compactification, the recent developments on log stable maps provide us a powerful tool to study A 1 -connectedness. We refer to [Kat89] for the basics of logarithmic geometry, and to [GS13,Che14,AC14,ACMW14,Wis14] for the details of the theory of stable log maps.…”

Skeletons and Fans of Logarithmic Structures

“…The boundedness of M Γ (X) has been established when the characteristic monoid M X is globally generated in [AC14,Che14], and more generally when the associated group M gp X is globally generated in [GS13]. The strategy used in [ACMW] for the general setting is to reduce to the case of a globally generated sheaf of monoids M X by studying the behavior of stable logarithmic maps under an appropriate modification of X.…”

“…Let X be a projective logarithmically smooth variety over C, and denote by M(X) the moduli space of logarithmic stable maps to X, as defined in [GS13,Che14,AC14]. Fix a logarithmicalyétale morphism h : Y → X of projective logarithmically smooth varieties.…”