177 research outputs found

    Symmetries of Riemann surfaces and magnetic monopoles

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    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 P3\mathbb P^3

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    Geproci sets of points in P3\mathbb P^3 are sets whose general projections to P2\mathbb P^2 are complete intersections. The first nontrivial geproci sets came from representation theory, as projectivizations of the root systems D4D_4 and F4F_4. 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 (a,b)(a,b)-geproci set ZZ with dd being the least degree of a space curve CC containing ZZ, that if dbd\leq b, then CC is a union of skew lines and ZZ 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 aa points on each of bb skew lines when ab13a\geq b-1\geq 3. We then introduce a notion of combinatorics for skew lines and apply it to the classification of single orbit combinatorial half grids of mm points on each of 4 lines. We apply these results to prove Theorem C, showing, when nmn\gg m, that half grids of mm points on nn 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 (m21)/12(m^2-1)/12 equivalence classes of combinatorial [m,4][m,4]-half grids in the two transversal case when m>2m>2 is prime.Comment: 36 page

    Characteristic polynomials and eigenvalues of tensors

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    We lay the geometric foundations for the study of the characteristic polynomial of tensors. For symmetric tensors of order d3d \geq 3 and dimension 22 and symmetric tensors of order 33 and dimension 33, 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

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    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 Mˉ0,n\bar{M}_{0,n}. 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

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    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

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    In this thesis we treat two topics: the construction of minimal complex surfaces of general type with pg=q=2,3p_g=q=2,3 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 pg=qp_g=q developed together with Fabrizio Catanese in [AC22]. We give first a construction of minimal surfaces of general type with pg=q=2p_g=q=2, K2=5K^2=5 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 SS has Albanese surface Alb(S)Alb(S) 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 pg=q=2p_g=q=2, K2=6K^2=6 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 pg=qp_g=q whose Tschirnhaus module has a kernel realization with quotient a nontrivial homogeneous bundle. Two families have pg=q=3p_g=q=3 (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 pg=q=2p_g=q=2, K2=6K^2=6 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 KK. 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 KK 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 Λ{2},{3}\Lambda_{\{2\},\{3\}} on some arrangements of six lines

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    The operator Λ{2},{3}\Lambda_{\{2\},\{3\}} 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 Λ{2},{3}\Lambda_{\{2\},\{3\}} of an unassuming arrangement is again an unassuming arrangement. We study the dynamics of the operator Λ{2},{3}\Lambda_{\{2\},\{3\}} 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

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    Bott proved a strong vanishing theorem for sheaf cohomology on projective space, namely that Hj(X,ΩXiL)=0H^j(X,\Omega^i_X\otimes L)=0 for every j>0j>0, i0i\geq 0, and LL 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 π ⁣:XY\pi\colon X\to Y of smooth varieties, and every line bundle AA on XX that is ample over YY, the higher direct image sheaf Rjπ(ΩXiA)R^j\pi_*(\Omega^i_X\otimes A) is zero for every j>0j>0 and i0i\geq 0.Comment: 31 page

    The Calabi Problem for Fano Threefolds

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    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

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    We present an algorithm for constructing a map P2P2\mathbb{P}^2\to\mathbb{P}^2 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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