Construction Of Moduli Space Research Articles

Moduli spaces of complex affine and dilation surfaces

Abstract We construct moduli spaces of complex affine and dilation surfaces. Using ideas of Veech [W. A. Veech, Flat surfaces, Amer. J. Math. 115 1993, 3, 589–689], we show that the moduli space 𝒜 g , n ⁢ ( 𝒎 ) {{\mathcal{A}}_{g,n}(\boldsymbol{m})} of genus g affine surfaces with cone points of complex order 𝒎 = ( m 1 ⁢ … , m n ) {\boldsymbol{m}=(m_{1}\ldots,m_{n})} is a holomorphic affine bundle over ℳ g , n {\mathcal{M}_{g,n}} , and the moduli space 𝒟 g , n ⁢ ( 𝒎 ) {{\mathcal{D}}_{g,n}(\boldsymbol{m})} of dilation surfaces is a covering space of ℳ g , n {\mathcal{M}_{g,n}} . We then classify the connected components of 𝒟 g , n ⁢ ( 𝒎 ) {{\mathcal{D}}_{g,n}(\boldsymbol{m})} and show that it is an orbifold- K ⁢ ( G , 1 ) {K(G,1)} , where G is the framed mapping class group of [A. Calderon and N. Salter, Framed mapping class groups and the monodromy of strata of Abelian differentials, preprint 2020].

A general notion of coherent systems

We look at coherent systems for decorated vector bundles and propose a notion of semistability. In the special case of tensor powers, we will examine this notion more closely. In particular, we will construct moduli spaces with the help of geometric invariant theory. It is an interesting aspect that ampleness of the linearization in the geometric invariant theory construction yields a bound on the stability parameter for coherent systems.

Toric construction and Chow ring of moduli space of quasi maps from ℙ1 with two marked points to ℙ1 × ℙ1

In this paper, we present explicit toric construction of moduli space of quasi maps from [Formula: see text] with two marked points to [Formula: see text], which was first proposed by Jinzenji and prove that it is a compact orbifold. We also determine its Chow ring and compute its Poincaré polynomial for some lower degree cases.

Moduli spaces of twisted K3 surfaces and cubic fourfolds

Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order. We do this by mimicking the construction of the moduli space of untwisted polarized K3 surfaces as a quotient of a bounded symmetric domain.

Monads for framed sheaves on Hirzebruch surfaces

Abstract We define monads for framed torsion-free sheaves on Hirzebruch surfaces and use them to construct moduli spaces for these objects. These moduli spaces are smooth algebraic varieties, and we show that they are fine by constructing a universal monad.

Instantons and vortices on noncommutative toric varieties

We elaborate on the quantization of toric varieties by combining techniques from toric geometry, isospectral deformations and noncommutative geometry in braided monoidal categories, and the construction of instantons thereon by combining methods from noncommutative algebraic geometry and a quantized twistor theory. We classify the real structures on a toric noncommutative deformation of the Klein quadric and use this to derive a new noncommutative four-sphere which is the unique deformation compatible with the noncommutative twistor correspondence. We extend the computation of equivariant instanton partition functions to noncommutative gauge theories with both adjoint and fundamental matter fields, finding agreement with the classical results in all instances. We construct moduli spaces of noncommutative vortices from the moduli of invariant instantons, and derive corresponding equivariant partition functions which also agree with those of the classical limit.

GIT stability of weighted pointed curves

In the late 1970s Mumford established Chow stability of smooth unpointed genus g curves embedded by complete linear systems of degree d ≥ 2g + 1, and at about the same time Gieseker established asymptotic Hilbert stability (that is, stability of m th Hilbert points for some large values of m) under the same hypotheses. Both of them then use an indirect argument to show that nodal Deligne-Mumford stable curves are GIT stable. The case of marked points lay untouched until 2006, when Elizabeth Baldwin proved that pointed Deligne-Mumford stable curves are asymptotically Hilbert stable. (Actually, she proved this for stable maps, which includes stable curves as a special case.) Her argument is a delicate induction on g and the number of marked points n; elliptic tails are glued to the marked points one by one, ultimately relating stability of an n-pointed genus g curve to Gieseker’s result for genus g + n unpointed curves. There are three ways one might wish to improve upon Baldwin’s results. First, one might wish to construct moduli spaces of weighted pointed curves or maps; it appears that Baldwin’s proof can accommodate some, but not all, sets of weights. Second, one might wish to study Hilbert stability for small values of m; since Baldwin’s proof uses Gieseker’s proof as the base case, it is not easy to see how it could be modified to yield an approach for small m. Finally, the Minimal Model Program for moduli spaces of curves has generated interest in GIT for 2, 3, or 4-canonical linear systems; due to its use of elliptic tails, Baldwin’s proof cannot be used to study these, as elliptic tails are known to be GIT unstable in these cases. In this paper I give a direct proof that smooth curves with distinct weighted marked points are asymptotically Hilbert stable with respect to a wide range of parameter spaces and linearizations. Some of these yield the (coarse) moduli space of Deligne-Mumford stable pointed curves M g,n and Hassett’s moduli spaces of weighted pointed curves M g,A, while other linearizations may give other quotients which are birational to these and which may admit interpretations as moduli spaces. The full construction of the moduli spaces is not contained in this paper, only the proof that smooth curves with distinct weighted marked points are stable, which is the key new result needed for the construction. For this I follow Gieseker’s approach to reduce to the GIT problem to a combinatorial problem, though the solution is very different.

Vector Bundles Near Negative Curves: Moduli and Local Euler Characteristic

We study moduli of vector bundles on a two-dimensional neighbourhood Z k of an irreducible curve ℓ ≅ ℙ1 with ℓ2 = −k and give an explicit construction of their moduli stacks. For the case of instanton bundles, we stratify the stacks and construct moduli spaces. We give sharp bounds for the local holomorphic Euler characteristic of bundles on Z k and prove existence of families of bundles with prescribed numerical invariants. Our numerical calculations are performed using a Macaulay 2 algorithm, which is available for download at http://www.maths.ed.ac.uk/~s0571100/Instanton/.

Abelianisation of orthogonal groups and the fundamental group of modular varieties

We study the commutator subgroup of integral orthogonal groups belonging to indefinite quadratic forms. We show that the index of this commutator is 2 for many groups that occur in the construction of moduli spaces in algebraic geometry, in particular the moduli of K3 surfaces. We give applications to modular forms and to computing the fundamental groups of some moduli spaces.

EXISTENCE OF COHERENT SYSTEMS II

A coherent system of type (r, d, k) on a curve consists of a vector bundle of rank r and degree d together with a vector space of dimension k of the sections of this vector bundle. There is a stability condition depending on a positive real parameter α that allows to construct moduli spaces for these objects. This paper shows non-emptiness of these moduli spaces when k > r for any α under some mild conditions on the degree and genus.

On the moduli of surfaces admitting genus two fibrations over elliptic curves

In this paper, we study the structure, deformations and the moduli spaces of complex projective surfaces admitting genus two fibrations over elliptic curves. We observe that a surface admitting a smooth fibration as above is elliptic, and we employ results on the moduli of polarized elliptic surfaces to construct moduli spaces of these smooth fibrations. In the case of nonsmooth fibrations, we relate the moduli spaces to the Hurwitz schemes \({\mathcal{H}}(1,X(d), n)\) of morphisms of degree n from elliptic curves to the modular curve X(d), d ≥ 3. Ultimately, we show that the moduli spaces in the nonsmooth case are fiber spaces over the affine line \({\mathbb{A}}^1\) with fibers determined by the components of \({\mathcal{H}}(1,X(d), n)\).

Non-linear Yang–Mills instantons from strings are π-stable D-branes

We show that B-type Π-stable D-branes do not in general reduce to the (Gieseker-) stable holomorphic vector bundles used in mathematics to construct moduli spaces. We show that solutions of the almost Hermitian Yang–Mills equations for the non-linear deformations of Yang–Mills instantons that appear in the low-energy geometric limit of strings exist iff they are π-stable, a geometric large volume version of Π-stability. This shows that π-stability is the correct physical stability concept. We speculate that this string-canonical choice of stable objects, which is encoded in and derived from the central charge of the string- algebra, should find applications to algebraic geometry where there is no canonical choice of stable geometrical objects.

Moduli for equivariant vector bundles of rank two on smooth toric surfaces

AbstractWe give a complete classification of equivariant vector bundles of rank two over smooth complete toric surfaces and construct moduli spaces of such bundles. This note is a direct continuation of an earlier note where we developed a general description of equivariant sheaves on toric varieties. Here we give a first application of that description. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)