arXiv subjects

Kenneth Ascher

Publications and source records attributed to Kenneth Ascher.

25 records · Page 2Linked to original sources

arXiv2017

Bounding heights uniformly in families of hyperbolic varieties

We show that, assuming Vojta's height conjecture, the height of a rational point on an algebraically hyperbolic variety can be bounded "uniformly" in families. This generalizes a result of Su-Ion Ih for curves of genus at least two to higher-dimensional varieties. As an application, we show that, assuming Vojta's height conjecture, the height of a rational point on a curve of general type is uniformly bounded. Finally, we prove a similar result for smooth hyperbolic surfaces with $c_1^2 > c_2$.

math.AG↗

arXiv2017

Log canonical models of elliptic surfaces

We give a classification of the log canonical models of elliptic surface pairs consisting of an elliptic fibration, a section, and a weighted sum of marked fibers. In particular, we show how the log canonical models depend on the choice of the weights. We describe a wall and chamber decomposition of the space of weights based on how the log canonical model changes. In addition, we give a generalized formula for the canonical bundle of an elliptic surface with section and marked fibers. This is the first step in constructing compactifcations of moduli spaces of elliptic surfaces using the minimal model program.

math.AG↗

arXiv2017

A Generic Slice of the Moduli Space of Line Arrangements

We study the compactification of the locus parametrizing lines with a fixed intersection to a given line, inside the moduli space of line arrangements in the projective plane constructed for weight one by Hacking-Keel-Tevelev and Alexeev for general weights. We show that this space is smooth, with normal crossing boundary, and that it has a morphism to the moduli space of marked rational curves which can be understood as a natural continuation of the blow up construction of Kapranov. In addition, we prove that it is isomorphic to a closed subvariety inside a non-reductive Chow quotient. The parametrized objects are surfaces with broken lines, whose dual graphs are rooted trees with possibly repeated markings.

math.AG↗

arXiv2016

Moduli of fibered surface pairs from twisted stable maps

We use the theory of twisted stable maps to Deligne-Mumford stacks to construct compactifications of the moduli space of pairs $(X \to C, S + F)$ where $X \to C$ is a fibered surface, $S$ is a sum of sections, $F$ is a sum of marked fibers, and $(X, S + F)$ is a stable pair in the sense of the minimal model program. This generalizes the work of Abramovich-Vistoli, who compactified the moduli space of fibered surfaces without marked fibers. Furthermore, we compare our compactification to Alexeev's space of stable maps and the KSBA compactification of stable pairs. As an application, we describe the boundary of the compactification of the moduli space of elliptic surfaces.

math.AG↗

arXiv2016

A fibered power theorem for pairs of log general type

Let $f: (X,D) \to B$ be a stably family with log canonical general fiber. We prove that, after a birational modification of the base $\tilde{B} \to B$, there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphism from a high fibered power of the original family to a pair openly of log general type.

math.AG↗

arXiv2016

Rational points on twisted K3 surfaces and derived equivalences

Using a construction of Hassett--Várilly-Alvarado, we produce derived equivalent twisted K3 surfaces over $\mathbb{Q}$, $\mathbb{Q}_2$, and $\mathbb{R}$, where one has a rational point and the other does not. This answers negatively a question recently raised by Hassett and Tschinkel.

math.NT↗

arXiv2015

Logarithmic stable toric varieties and their moduli

The Chow quotient of a toric variety by a subtorus, as defined by Kapranov-Sturmfels-Zelevinsky, coarsely represents the main component of the moduli space of stable toric varieties with a map to a fixed projective toric variety, as constructed by Alexeev and Brion. We show that, after endowing both spaces with the structure of a logarithmic stack, the resulting spaces are isomorphic. Along the way, we construct the Chow quotient stack and demonstrate several properties that it satisfies.

math.AG↗