177 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
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
Characteristic polynomials and eigenvalues of tensors
We lay the geometric foundations for the study of the characteristic
polynomial of tensors. For symmetric tensors of order and dimension
and symmetric tensors of order and dimension , we prove that only
finitely many tensors share any given characteristic polynomial, unlike the
case of symmetric matrices and the case of non-symmetric tensors. We propose
precise conjectures for the dimension of the variety of tensors sharing the
same characteristic polynomial, in the symmetric and in the non-symmetric
setting.Comment: 25 pages, comments are welcom
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, Birkhäuser.
[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, Birkhäuser.
[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. "Über die Darstellung der endlichen Gruppen durch gebrochen lineare Substitutionen". J. Reine Angew. Math. 127 (1904), 20--50 (German)
On the dynamics of the line operator on some arrangements of six lines
The operator acting on line arrangements is defined
by associating to a line arrangement \mathcal{A}, the line arrangement which is
the union of the lines containing exactly three points among the double points
of \mathcal{A}. We say that six lines not tangent to a conic form an unassuming
arrangement if the singularities of their union are only double points, but the
dual line arrangement has six triple points, six 5-points and 27 double points.
The moduli space of unassuming arrangements is the union of a point and a line.
The image by the operator of an unassuming arrangement
is again an unassuming arrangement. We study the dynamics of the operator
on these arrangements and we obtain that the periodic
arrangements are related to the Ceva arrangements of lines.Comment: 20 pages, comments welcom
Bott vanishing for Fano 3-folds
Bott proved a strong vanishing theorem for sheaf cohomology on projective
space, namely that for every , ,
and ample. This holds for toric varieties, but not for most other
varieties.
We classify the smooth Fano 3-folds that satisfy Bott vanishing. There are
many more than expected.
Along the way, we conjecture that for every projective birational morphism
of smooth varieties, and every line bundle on that
is ample over , the higher direct image sheaf is zero for every and .Comment: 31 page
The Calabi Problem for Fano Threefolds
There are 105 irreducible families of smooth Fano threefolds, which have been classified by Iskovskikh, Mori and Mukai. For each family, we determine whether the general member admits a Kähler–Einstein metric or not. We also find all Kähler–Einstein smooth Fano threefolds that have infinite automorphism groups
How to Reconstruct a Planar Map From its Branching Curve
We present an algorithm for constructing a map
with a given branching curve. The stepping stone is the ramification curve,
which is obtained as the linear normalization of the branching curve
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