490 research outputs found
Symmetries of Riemann surfaces and magnetic monopoles
This thesis studies, broadly, the role of symmetry in elucidating structure. In particular, I investigate the role that automorphisms of algebraic curves play in three specific contexts; determining the orbits of theta characteristics, influencing the geometry of the highly-symmetric Bring’s curve, and in constructing magnetic monopole solutions. On theta characteristics, I show how to turn questions on the existence of invariant characteristics into questions of group cohomology, compute comprehensive tables of orbit decompositions for curves of genus 9 or less, and prove results on the existence of infinite families of curves with invariant characteristics. On Bring’s curve, I identify key points with geometric significance on the curve, completely determine the structure of the quotients by subgroups of automorphisms, finding new elliptic curves in the process, and identify the unique invariant theta characteristic on the curve. With respect to monopoles, I elucidate the role that the Hitchin conditions play in determining monopole spectral curves, the relation between these conditions and the automorphism group of the curve, and I develop the theory of computing Nahm data of symmetric monopoles. As such I classify all 3-monopoles whose Nahm data may be solved for in terms of elliptic functions
Higher Chow cycles on some K3 surfaces with involution
We construct, for each 2<r<18, an explicit family of higher Chow cycles of
type (2,1) on a family of lattice-polarized K3 surfaces of generic Picard rank
r, and prove that the indecomposable part of this cycle is non-torsion for very
general members of the family. These are the first explicit examples of such
families in middle Picard rank. Our construction is based on singular double
plane model of K3 surfaces, and the proof of indecomposability is done by a
degeneration method
Hecke cycles on moduli of vector bundles and orbital degeneracy loci
Given a smooth genus two curve , the moduli space SU of rank three
semi-stable vector bundles on with trivial determinant is a double cover in
branched over a sextic hypersurface, whose projective dual is
the famous Coble cubic, the unique cubic hypersurface that is singular along
the Jacobian of . In this paper we continue our exploration of the
connections of such moduli spaces with the representation theory of ,
initiated in \cite{GSW} and pursued in \cite{GS, sam-rains1, sam-rains2, bmt}.
Starting from a general trivector in , we construct a
Fano manifold in as a so-called orbital degeneracy
locus, and we prove that it defines a family of Hecke lines in SU. We
deduce that is isomorphic to the odd moduli space SU of rank three stable vector bundles on with fixed
effective determinant of degree one. We deduce that the intersection of
with a general translate of in is a K3
surface of genus
New perspectives on categorical Torelli theorems for del Pezzo threefolds
Let be a del Pezzo threefold of Picard rank one and degree .
In this paper, we apply two different viewpoints to study via a
particular admissible subcategory of its bounded derived category, called the
Kuznetsov component:
(i) Brill-Noether reconstruction. We show that can be uniquely
recovered as a Brill-Noether locus of Bridgeland stable objects in its
Kuznetsov component.
(ii) Exact equivalences. We prove that, up to composing with an explicit
auto-equivalence, any Fourier-Mukai type equivalence of Kuznetsov components of
two del Pezzo threefolds of degree can be lifted to an
equivalence of their bounded derived categories. As a result, we obtain a
complete description of the group of Fourier-Mukai type auto-equivalences of
the Kuznetsov component of .
We also describe the group of Fourier-Mukai type auto-equivalences of
Kuznetsov components of index one prime Fano threefolds of genus
and . As an application, first we identify the group of automorphisms
of and its associated . Then we give a new disproof of
Kuznetsov's Fano threefold conjecture by assuming Gushel-Mukai threefolds are
general.
In an appendix, we classify instanton sheaves on quartic double solids,
generalizing a result of Druel.Comment: 35 pages, added results on index one prime Fano threefolds and the
application to Kuznetsov's Fano threefold conjecture. Comments are very
welcome
Splitting invariants and a π1-equivalent Zariski-pair of conic-line arrangements
於 Zoom (2022年10月18日-10月21日)2022年度科学研究費補助金 基盤研究(A)(課題番号 21H04429, 代表 並河良典)世話人: 池田京司(東京電機大), 稲場道明(京都大), 深澤知(山形大)This article is based on joint work with M. Amram (Shamoon College of Engineering, Israel), T. Shirane (Tokushima U.), U. Sinichkin (Tel Aviv University, Israel) and H. Tokunaga (TMU).This article is based on the authors talk given at the Kinosaki Algebraic Geometry Symposium 2022. We give a brief overview of the subject of the embedded topology of plane curves. Furthermore, we illustrate the idea of a relatively new type of invariants called splitting invariants which prove effective in distinguishing the topology of plane curves. We also describe a new example of a π1-equivalent Zariski-pair consisting of conic-line arrangements of degree 7
New sextics of genus 6 and 10 attaining the Serre bound
We provide new examples of curves of genus 6 or 10 attaining the Serre bound.
They all belong to the family of sextics introduced in [19] as a a
generalization of the Wiman sextics [36] and Edge sextics [9]. Our approach is
based on a theorem by Kani and Rosen which allows, under certain assumptions,
to fully decompose the Jacobian of the curve. With our investigation we are
able to update several entries in \url{http://www.manypoints.org} ([35])
Effective divisors on projectivized Hodge bundles and modular forms
We construct vector-valued modular forms on moduli spaces of curves and
abelian varieties using effective divisors in projectivized Hodge bundles over
moduli of curves. Cycle relations tell us the weight of these modular forms. In
particular we construct basic modular forms for genus and . We also
discuss modular forms on the moduli of hyperelliptic curves. In that case the
relative canonical bundle is a pull back of a line bundle on a -bundle over the moduli of hyperelliptic curves and we extend that line
bundle to a compactification so that its push down is (close to) the Hodge
bundle and use this to construct modular forms. In an appendix we use our
method to calculate divisor classes in the dual projectivized -Hodge bundle
determined by Gheorghita-Tarasca and by Korotkin-Sauvaget-Zograf.Comment: 31 pages; to appear in Math. Nachrichte
Old and new motivic cycles on Abelian surfaces
Collino \cite{colo} discovered indecomposable motivic cycles in the group
H^{2g-1}_{\mathcal M}(J(C),{\mathds Z}(g)). In an earlier paper we described
the construction of some new motivic cycles which can be viewed as a
generalization of Collino's cycle when . In this paper we show that our
new cycles are in fact related to Collino's cycles of higher genus. On one hand
this suggests that new cycles are hard to find. On the other, it suggests that
the tools developed to study Collino's cycle can be applied to our cycles.Comment: 13 page
Extremal Cubic Inequalities of Three Variables
Let be the vector space of homogeneous three variable
polynomials of degree , and be the set of all elements
such that for all , , . In this article, we determine all extremal elements of
. We prove that if is an
irreducible extremal element, then the zero locus in
is a rational curve whose singularity is an acnode
in the interior of or a cusp on an edge of .
We also prove that if is an extremal element, then
is an extremal element of , where
is the set of all the elements
such that for all , , . A notion of
infinitely near zeros of an inequality is introduced, and plays an important
role
The realization space of a certain conic line arrangement of degree 7 and a -equivalent Zariski pair
In this paper, we continue the study of the embedded topology of plane
algebraic curves. We study the realization space of conic line arrangements of
degree with certain fixed combinatorics and determine the number of
connected components. This is done by showing the existence of a Zariski pair
having these combinatorics, which we identified as a -equivalent Zariski
pair.Comment: 24 page
- …