163 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 Bringâs 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 Bringâs 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
Geproci sets and the combinatorics of skew lines in
Geproci sets of points in are sets whose general projections to
are complete intersections. The first nontrivial geproci sets
came from representation theory, as projectivizations of the root systems
and . In most currently known cases geproci sets lie on very special
unions of skew lines and are known as half grids. For this important class of
geproci sets we establish fundamental connections with combinatorics, which we
study using methods of algebraic geometry and commutative algebra. As a
motivation for studying them, we first prove Theorem A: for a nondegenerate
-geproci set with being the least degree of a space curve
containing , that if , then is a union of skew lines and is
either a grid or a half grid. We next formulate a combinatorial version of the
geproci property for half grids and prove Theorem B: combinatorial half grids
are geproci in the case of sets of points on each of skew lines when
. We then introduce a notion of combinatorics for skew lines
and apply it to the classification of single orbit combinatorial half grids of
points on each of 4 lines. We apply these results to prove Theorem C,
showing, when , that half grids of points on lines with two
transversals must be very special geometrically (if they even exist). Moreover,
in the case of skew lines having two transversals, our results provide an
algorithm for enumerating their projective equivalence classes. We conjecture
there are equivalence classes of combinatorial -half grids
in the two transversal case when is prime.Comment: 36 page
Tropical invariants for binary quintics and reduction types of Picard curves
In this paper, we express the reduction types of Picard curves in terms of
tropical invariants associated to binary quintics. These invariants are
connected to Picard modular forms using recent work by Cl{\'e}ry and van der
Geer. We furthermore give a general framework for tropical invariants
associated to group actions on arbitrary varieties. The previous problem fits
in this general framework by mapping the space of binary forms to symmetrized
versions of the Deligne--Mumford compactification . We
conjecture that the techniques introduced here can be used to find tropical
invariants for binary forms of any degree
Computation and Physics in Algebraic Geometry
Physics provides new, tantalizing problems that we solve by developing and implementing innovative and effective geometric tools in nonlinear algebra. The techniques we employ also rely on numerical and symbolic computations performed with computer algebra.
First, we study solutions to the Kadomtsev-Petviashvili equation that arise from singular curves. The Kadomtsev-Petviashvili equation is a partial differential equation describing nonlinear wave motion whose solutions can be built from an algebraic curve. Such a surprising connection established by Krichever and Shiota also led to an entirely new point of view on a classical problem in algebraic geometry known as the Schottky problem. To explore the connection with curves with at worst nodal singularities, we define the Hirota variety, which parameterizes KP solutions arising from such curves. Studying the geometry of the Hirota variety provides a new approach to the Schottky problem. We investigate it for irreducible rational nodal curves, giving a partial solution to the weak Schottky problem in this case.
Second, we formulate questions from scattering amplitudes in a broader context using very affine varieties and D-module theory. The interplay between geometry and combinatorics in particle physics indeed suggests an underlying, coherent mathematical structure behind the study of particle interactions. In this thesis, we gain a better understanding of mathematical objects, such as moduli spaces of point configurations and generalized Euler integrals, for which particle physics provides concrete, non-trivial examples, and we prove some conjectures stated in the physics literature.
Finally, we study linear spaces of symmetric matrices, addressing questions motivated by algebraic statistics, optimization, and enumerative geometry. This includes giving explicit formulas for the maximum likelihood degree and studying tangency problems for quadric surfaces in projective space from the point of view of real algebraic geometry
Constructions and moduli of surfaces of general type and related topics
In this thesis we treat two topics: the construction of minimal complex surfaces of general type with and an extension of Schur's concept of a representation group for projective representations to the setting of semi-projective representations.
These are the contents of the two articles [AC22] and [AGK23], which are two joint works: the former with Fabrizio Catanese, the latter with Christian Gleissner and Julia Kotonski.
The first part of the thesis is devoted to the treatment of the construction method for minimal surfaces of general type with developed together with Fabrizio Catanese in [AC22].
We give first a construction of minimal surfaces of general type with , and Albanese map of degree 3, describing a unirational irreducible connected component of the Gieseker moduli space, which we show to be the only one with these invariants fulfilling a mild technical assumption (which we call Gorenstein Assumption) and whose general element has Albanese surface containing no elliptic curve.
We call it the component of "CHPP surfaces", since it contains the family constructed by Chen and Hacon in [CH06], and coincides with the one constructed by Penegini and Polizzi in [PePo13a].
Similarly, we construct a unirational irreducible connected component of the moduli space of minimal surfaces of general type with , and Albanese map of degree 4, which we call the component of "PP4 surfaces" since it coincides with the irreducible one constructed by Penegini and Polizzi in [PePo14].
Furthermore, we answer a question posed by Chen and Hacon in [CH06] by constructing three families of surfaces with whose Tschirnhaus module has a kernel realization with quotient a nontrivial homogeneous bundle.
Two families have (one of them is just a potential example since a computer script showing the existence is still missing), while the third one is a new family of surfaces with , and Albanese map of degree 3.
The latter, whose existence is showed in [CS22], yields a new irreducible component of the Gieseker moduli space, which we call the component of "AC3 surfaces". This is the first known component with these invariants, and moreover we show that it is unirational.
We point out that we provide explicit and global equations for all the five families of surfaces we mentioned above.
Finally, in the second and last part of the thesis we treat the content of the joint work [AGK23] with Christian Gleissner and Julia Kotonski.
Here we study "semi-projective representations", i.e., homomorphisms of finite groups to the group of semi-projective transformations of finite dimensional vector spaces over an arbitrary field . The main tool we use is "group cohomology", more precisely explicit computations involving cocycles.
As our main result, we extend Schur's concept of "projective representation groups" [Sch04] to the semi-projective case under the assumption that is algebraically closed.
Furthermore, a computer algorithm is given: it produces, for a given finite group, all "twisted representation groups" under trivial or conjugation actions on the field of complex numbers.
In order to stress the relevance of the theory, we discuss two important applications, where semi-projective representations occur naturally.
The first one reviews Isaacs' treatment in "Clifford theory for characters" [Isa81], namely the extension problem of invariant characters (over arbitrary fields) defined on normal subgroups.
The second one is our original algebro-geometric motivation and deals with the problem to find linear parts of homeomorphisms and biholomorphisms between complex torus quotients.
References:
[AC22] Massimiliano Alessandro and Fabrizio Catanese. "On the components of the Main Stream of the moduli space of surfaces of general type with p_g=q=2". Preprint 2022 (arXiv:2212.14872v3). To appear in "Perspectives on four decades: Algebraic Geometry 1980-2020. In memory of Alberto Collino". Trends in Mathematics, BirkhÀuser.
[AGK23] Massimiliano Alessandro, Christian Gleissner and Julia Kotonski. "Semi-projective representations and twisted representation groups". Comm. Algebra 51 (2023), no. 10, 4471--4480.
[CS22] Fabrizio Catanese and Edoardo Sernesi. "The Hesse pencil and polarizations of type (1,3) on abelian surfaces". Preprint 2022 (arXiv:2212.14877). To appear in "Perspectives on four decades: Algebraic Geometry 1980-2020. In memory of Alberto Collino". Trends in Mathematics, BirkhÀuser.
[CH06] Jungkai Alfred Chen and Christopher Derek Hacon. "A surface of general type with p_g=q=2 and K^2=5". Pacific J. Math. 223 (2006), no. 2, 219--228.
[Isa81] Irving Martin Isaacs. "Extensions of group representations over arbitrary fields". J. Algebra 68 (1981), no. 1, 54--74.
[PePo13a] Matteo Penegini and Francesco Polizzi. "On surfaces with p_g=q=2, K^2=5 and Albanese map of degree 3". Osaka J. Math. 50 (2013), no. 3, 643--686.
[PePo14] Matteo Penegini and Francesco Polizzi. "A new family of surfaces with p_g=q=2 and K^2=6 whose Albanese map has degree 4". J. Lond. Math. Soc. (2) 90 (2014), no. 3, 741--762.
[Sch04] Issai Schur. "Ăber die Darstellung der endlichen Gruppen durch gebrochen lineare Substitutionen". J. Reine Angew. Math. 127 (1904), 20--50 (German)
Birational geometry of moduli space of del Pezzo pairs
In this paper, we investigate the geometry of moduli space of degree
del Pezzo pair, that is, a del Pezzo surface of degree with a curve
. More precisely, we study compactifications for from both
Hodge's theoretical and geometric invariant theoretical (GIT) perspective. We
compute the Picard numbers of these compact moduli spaces which is an important
step to set up the Hassett-Keel-Looijenga models for . For case, we
propose the Hassett-Keel-Looijenga program \cF_8(s)=\Proj(R(\cF_8,\Delta(s) )
as the section rings of certain \bQ-line bundle on locally
symmetric variety \cF_8, which is birational to . Moreover, we give an
arithmetic stratification on \cF_8. After using the arithmetic computation of
pullback on these arithmetic strata, we give the arithmetic
predictions for the wall-crossing behavior of \cF_8(s) when
varies. The relation of \cF_8(s) with the K-moduli spaces of degree del
Pezzo pairs is also proposed.Comment: 43 pages, comments are very welcome
K-moduli of log Fano_complete_intersections
We explicitly describe the K-moduli compactifications and wall-crossings of log pairs formed by a Fano complete intersection of two quadric threefolds and a hyperplane, by constructing an isomorphism with the VGIT quotient of such complete intersections and a hyperplane. We further characterize all possible such GIT quotients based on singularities. Our main result is the first example of wall-crossing for the K-moduli of log pairs, where both the variety and divisor admit deformations before and after the wall-crossing. Furthermore, we explicitly describe the K-moduli of the deformation family of Fano 3-folds 2.25 in the Mori-Mukai classification, which can be viewed as blow ups of complete intersections of two quadrics in dimension three, by showing there exists an isomorphism to a GIT quotient which we also explicitly describe. This is one of the only three known explicit compactifications of K-moduli of Fano threefolds.
Our work uses the moduli continuity method for log pairs by relating the K-moduli to certain GIT compactifications. In addition, we introduce the reverse moduli continuity method, which allows us to relate canonical GIT compactifications to K-moduli of Fano varieties. We also compute the CM line bundle for complete intersections of hypersurfaces of fixed degree with a hyperplane section, and we show it isomorphic to an ample line bundle in the Picard group of the canonical GIT quotient of complete intersections and a hyperplane. We use explicit GIT methods to classify in detail the GIT stability of a complete intersection of two quadrics in dimension four and three, together with a hyperplane section. Furthermore, we explicitly compactify the moduli space of log Fano pairs of complete intersections and hyperplane sections, by establishing a direct link with the GIT compactification
Parameter estimation with gravitational waves
The new era of gravitational wave astronomy truly began on September 14, 2015
with the detection of GW150914, the sensational first direct observation of
gravitational waves from the inspiral and merger of two black holes by the two
Advanced LIGO detectors. In the subsequent first three observing runs of the
LIGO/Virgo network, gravitational waves from compact binary mergers
have been announced, with more results to come. The events have mostly been
produced by binary black holes, but two binary neutron star mergers have so far
been observed, as well as the mergers of two neutron star - black hole systems.
Furthermore, gravitational waves emitted by core-collapse supernovae, pulsars
and the stochastic gravitational wave background are within the
LIGO/Virgo/KAGRA sensitivity band and are likely to be observed in future
observation runs. Beyond signal detection, a major challenge has been the
development of statistical and computational methodology for estimating the
physical waveform parameters and quantifying their uncertainties in order to
accurately characterise the emitting system. These methods depend on the
sources of the gravitational waves and the gravitational waveform model that is
used. This article reviews the main waveform models and parameter estimation
methods used to extract physical parameters from gravitational wave signals
detected to date by LIGO and Virgo and from those expected to be observed in
the future, which will include KAGRA, and how these methods interface with
various aspects of LIGO/Virgo/KAGRA science. Also presented are the statistical
methods used by LIGO and Virgo to estimate detector noise, test general
relativity, and draw conclusions about the rates of compact binary mergers in
the universe. Furthermore, a summary of major publicly available gravitational
wave parameter estimation software packages is given
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