Let M0,n(r, d) be the coarse moduli space of stable maps of degree d from npointed genus 0 nodal curves to Pr. Let B0,n(r, d) be the coarse moduli space of n-pointed branchvarieties of Pr with reduced and connected 1-dimensional domain and fixed Hilbert polynomial h(m) = dm+1. In [AL06], we prove that there is a surjective birational (small) morphism φ : M0,n(r, d) −→ B0,n(r, d). In this paper,...

Let φ : X → X be a morphism of a variety over a number field K. We consider local conditions and a “Brauer-Manin” condition, defined by Hsia and Silverman, for the orbit of a point P ∈ X(K) to be disjoint from a subvariety V ⊆ X, i.e., for Oφ(P )∩ V = ∅. We provide evidence that the dynamical Brauer-Manin condition is sufficient to explain the lack of points in the intersection Oφ(P ) ∩ V ; thi...

We prove the termination of 4-fold canonical flips. This paper is a supplement of [KMM, Theorem 5-1-15] and [M, Main Theorem 2.1]. We prove that a sequence of 4-fold log flips for canonical pairs terminates after finitely many steps. Let us recall the definition of canonical flips, which is slightly different from the usual one (cf. [S, (2.11) Adjoint Diagram]). Definition 1 (Canonical flip). L...

Throughout this paper, most our notations and terminologies are standard within algebraic geometry except for the following which we are in favour of: := — definition; ∼lin — linear equivalence; ∼num — numerical equivalence. Let X be a complex nonsingular projective variety of general type with dimension d(d ≥ 2). Suppose dimΦ|KX |(X) = 1, we usually say that the canonical system |KX | is compo...

Problems. Problem 1. Let Y be a finite type, separated k-scheme. Let E be a locally free OY -module of rank r + 1. Let πE : PY (E)→ Y, φ : π∗E∨ → O(1) be a universal pair of a morphism to Y together with an invertible quotient of the pullback of E∨ (to help calibrate conventions, this is covariant in E with respect to locally split monomorphisms of locally free sheaves). Recall in the proof of ...

It has been shown [6] that, within the McAlister inverse monoid [9], whose elements can be seen as overlapping one-dimensional tiles, the class of languages recognizable by finite monoids collapses compared with the class of languages definable in Monadic Second Order Logic (MSO). This paper aims at capturing the expressive power of the MSO definability of languages of tiles by means of a weake...