490 research outputs found

    Symmetries of Riemann surfaces and magnetic monopoles

    Get PDF
    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

    Full text link
    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

    Full text link
    Given a smooth genus two curve CC, the moduli space SUC(3)_C(3) of rank three semi-stable vector bundles on CC with trivial determinant is a double cover in P8\mathbb{P}^8 branched over a sextic hypersurface, whose projective dual is the famous Coble cubic, the unique cubic hypersurface that is singular along the Jacobian of CC. In this paper we continue our exploration of the connections of such moduli spaces with the representation theory of GL9GL_9, initiated in \cite{GSW} and pursued in \cite{GS, sam-rains1, sam-rains2, bmt}. Starting from a general trivector vv in 3C9\wedge^3\mathbb{C}^9, we construct a Fano manifold DZ10(v)D_{Z_{10}}(v) in G(3,9)G(3,9) as a so-called orbital degeneracy locus, and we prove that it defines a family of Hecke lines in SUC(3)_C(3). We deduce that DZ10(v)D_{Z_{10}}(v) is isomorphic to the odd moduli space SUC(3,OC(c))_C(3, \mathcal{O}_C(c)) of rank three stable vector bundles on CC with fixed effective determinant of degree one. We deduce that the intersection of DZ10(v)D_{Z_{10}}(v) with a general translate of G(3,7)G(3,7) in G(3,9)G(3,9) is a K3 surface of genus 1919

    New perspectives on categorical Torelli theorems for del Pezzo threefolds

    Full text link
    Let YdY_d be a del Pezzo threefold of Picard rank one and degree d2d\geq 2. In this paper, we apply two different viewpoints to study YdY_d via a particular admissible subcategory of its bounded derived category, called the Kuznetsov component: (i) Brill-Noether reconstruction. We show that YdY_d 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 2d42\leq d\leq 4 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 YdY_d. We also describe the group of Fourier-Mukai type auto-equivalences of Kuznetsov components of index one prime Fano threefolds X2g2X_{2g-2} of genus g=6g=6 and 88. As an application, first we identify the group of automorphisms of X14X_{14} and its associated Y3Y_3. 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

    Get PDF
    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

    Full text link
    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

    Full text link
    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 22 and 33. 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 P1{\mathbb P}^1-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 kk-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

    Full text link
    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 g=2g=2. 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

    Full text link
    Let H3,d\mathcal{H}_{3,d} be the vector space of homogeneous three variable polynomials of degree dd, and P3,d+\mathcal{P}_{3,d}^+ be the set of all elements fH3,df \in \mathcal{H}_{3,d} such thatf(x,y,z)0f(x,y,z) \geq 0 for all x0x \geq 0, y0y \geq 0, z0z \geq 0. In this article, we determine all extremal elements of P3,3+\mathcal{P}_{3,3}^+. We prove that if fP3,3+f \in \mathcal{P}_{3,3}^+ is an irreducible extremal element, then the zero locus VC(f)V_{\mathbb{C}}(f) in PC2\mathbb{P}_{\mathbb{C}}^2 is a rational curve whose singularity is an acnode in the interior of P+2\mathbb{P}_+^2 or a cusp on an edge of P+2\mathbb{P}_+^2. We also prove that if fP3,3+f \in \mathcal{P}_{3,3}^+ is an extremal element, then f(x2,y2,z2)f(x^2,y^2,z^2) is an extremal element of P3,6\mathcal{P}_{3,6}, where P3,d\mathcal{P}_{3,d} is the set of all the elements fH3,df \in \mathcal{H}_{3,d} such that f(x,y,z)0f(x,y,z) \geq 0 for all xx, yy, zRz \in \mathbb{R}. 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 π1\pi_1-equivalent Zariski pair

    Full text link
    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 77 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 π1\pi_1-equivalent Zariski pair.Comment: 24 page
    corecore