3,701 research outputs found
Stable twisted curves and their r-spin structures
The object of this paper is the notion of r-spin structure: a line bundle
whose r-th power is isomorphic to the canonical bundle. Over the moduli functor
M_g of smooth genus- curves, -spin structures form a finite torsor under
the group of r-torsion line bundles. Over the moduli functor Mbar_g of stable
curves, r-spin structures form an 'etale stack, but the finiteness and the
torsor structure are lost.
In the present work, we show how this bad picture can be definitely improved
simply by placing the problem in the category of Abramovich and Vistoli's
twisted curves. First, we find that within such category there exist several
different compactifications of M_g; each one corresponds to a different
multiindex \ell=(l0,l1,...) identifying a notion of stability: \ell-stability.
Then, we determine the suitable choices of \ell for which r-spin structures
form a finite torsor over the moduli of \ell-stable curves.Comment: 44 pages, revised version, to appear in Annales de l'Institut Fourie
On torsion in finitely presented groups
We give a uniform construction that, on input of a recursive presentation
of a group, outputs a recursive presentation of a torsion-free group,
isomorphic to whenever is itself torsion-free. We use this to re-obtain
a known result, the existence of a universal finitely presented torsion-free
group; one into which all finitely presented torsion-free groups embed. We
apply our techniques to show that recognising embeddability of finitely
presented groups is -hard, -hard, and lies in
. We also show that the sets of orders of torsion elements of
finitely presented groups are precisely the sets which are
closed under taking factors.Comment: 11 pages. This is the version submitted for publicatio
LG/CY correspondence: the state space isomorphism
We prove the classical mirror symmetry conjecture for the mirror pairs
constructed by Berglund, H\"ubsch, and Krawitz. Our main tool is a
cohomological LG/CY correspondence which provides a degree-preserving
isomorphism between the cohomology of finite quotients of Calabi-Yau
hypersurfaces inside a weighted projective space and the Fan-Jarvis-Ruan-Witten
state space of the associated Landau-Ginzburg singularity theory.Comment: 37 pages, 9 figure
Singularities of the moduli space of level curves
We describe the singular locus of the compactification of the moduli space
of curves of genus paired with an -torsion point in their
Jacobian. Generalising previous work for , we also describe the
sublocus of noncanonical singularities for any positive integer . For and , this allows us to provide a lifting result on pluricanonical
forms playing an essential role in the computation of the Kodaira dimension of
: for those values of , every pluricanonical form on the smooth
locus of the moduli space extends to a desingularisation of the compactified
moduli space.Comment: 37 pages, 9 figures, to appear in J Eur Math So
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