66 research outputs found
Computing heights on weighted projective spaces
In this note we extend the concept height on projective spaces to that of
weighted height on weighted projective spaces and show how such a height can be
computed. We prove some of the basic properties of the weighted height and show
how it can be used to study hyperelliptic curves over Q. Some examples are
provided from the weighted moduli space of binary sextics and octavics
Rational points in the moduli space of genus two
We build a database of genus 2 curves defined over which contains
all curves with minimal absolute height , all curves with moduli
height , and all curves with extra automorphisms in
standard form defined over with height .
For each isomorphism class in the database, an equation over its minimal field
of definition is provided, the automorphism group of the curve, Clebsch and
Igusa invariants. The distribution of rational points in the moduli space
for which the field of moduli is a field of definition is
discussed and some open problems are presented
The KSBA compactification for the moduli space of degree two K3 pairs
Inspired by the ideas of the minimal model program, Shepherd-Barron,
Koll\'ar, and Alexeev have constructed a geometric compactification for the
moduli space of surfaces of log general type. In this paper, we discuss one of
the simplest examples that fits into this framework: the case of pairs (X,H)
consisting of a degree two K3 surface X and an ample divisor H. Specifically,
we construct and describe explicitly a geometric compactification
for the moduli of degree two K3 pairs. This compactification has a natural
forgetful map to the Baily-Borel compactification of the moduli space of
degree two K3 surfaces. Using this map and the modular meaning of ,
we obtain a better understanding of the geometry of the standard
compactifications of .Comment: 45 pages, 4 figures, 2 table
On the Geometry of the Moduli Space of Real Binary Octics
The moduli space of smooth real binary octics has five connected components.
They parametrize the real binary octics whose defining equations have 0, 1,
..., 4 complex-conjugate pairs of roots respectively. We show that the
GIT-stable completion of each of these five components admits the structure of
an arithmetic real hyperbolic orbifold. The corresponding monodromy groups are,
up to commensurability, discrete hyperbolic reflection groups, and their
Vinberg diagrams are computed. We conclude with a simple proof that the moduli
space of GIT-stable real binary octics itself cannot be a real hyperbolic
orbifold.Comment: 23 page
On hyperelliptic curves of genus 3
We study the moduli space of genus 3 hyperelliptic curves via the weighted
projective space of binary octavics. This enables us to create a database of
all genus 3 hyperelliptic curves defined over , of weighted moduli
height
The Chow ring of the moduli space of degree quasi-polarized K3 surfaces
We study the Chow ring with rational coefficients of the moduli space
of quasi-polarized surfaces of degree . We find
generators, relations, and calculate the Chow Betti numbers. The highest
nonvanishing Chow group is . We
prove that the Chow ring consists of tautological classes and is isomorphic to
the even cohomology. The Chow ring is not generated by divisors and does not
satisfy duality with respect to the pairing into . In the appendix, we revisit Kirwan-Lee's calculation of the Poincar\'e
polynomial of .Comment: 40 pages, comments welcome
- β¦