Measurements of the total cross-section difference and the parameter CLL in pp scattering with longitudinally-polarized beam and target (original) (raw)
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manuscripta mathematica, 2008
We study several deformation functors associated to the normalization of a reduced curve singularity (X, 0) ⊂ (C n , 0). The main new results are explicit formulas, in terms of classical invariants of (X, 0), for the cotangent cohomology groups T i , i = 0, 1, 2, of these functors. Thus we obtain precise statements about smoothness and dimension of the corresponding local moduli spaces. We apply the results to obtain explicit formulas resp. estimates for the Ae-codimension of a parametrized curve singularity, where Ae denotes the Mather-Wall group of left-right equivalence.
Topics in deformation and moduli theory for singularities on curves and surfaces
The aim of this thesis is to contribute to the understanding of moduli of isolated singularities in dimension one and two. Historically, Riemann classified the possible conformal structures on a compact Riemann surface. In algebraic geometry the problem of moduli has gotten increasing attention. Local moduli of singularities is one aspect, and Zariski considered in [Zar73] this problem for plane curve singularities of the form x m + y m+1 . Later Laudal and Pfister took a systematic approach to this problem for plane curve singularities of quasihomogeneous type, see .
Scattering amplitudes of stable curves
Cornell University - arXiv, 2020
Hypertree divisors on the moduli space of stable rational curves were introduced by Castravet and Tevelev in [CT1]. Their equations appear as numerators of scattering amplitude forms for n particles in N = 4 Yang-Mills theory in the work of Arkani-Hamed, Bourjaily, Cachazo, Postnikov and Trnka [ABC + 1]. Rather than being a coincidence, this is just the tip of the iceberg of an exciting relation between algebraic geometry and high energy physics. We interpret leading singularities of scattering amplitudes of massless particles as probabilistic Brill-Noether theory: the study of statistics of images of n marked points under a random meromorphic function uniformly distributed with respect to the translationinvariant volume form of the Jacobian. We focus on the maximum helicity violating case, which leads to a beautiful physics-inspired geometry for various classes of complex algebraic curves: smooth, stable, hyperelliptic, real algebraic, etc.
Equisingular calculations for plane curve singularities
Journal of Symbolic Computation, 2007
We present an algorithm which, given a deformation with section of a reduced plane curve singularity, computes equations for the equisingularity stratum (that is, the µ-constant stratum in characteristic 0) in the parameter space of the deformation. The algorithm works for any, not necessarily reduced, parameter space and for algebroid curve singularities C defined over an algebraically closed field of characteristic 0 (or of characteristic p > ord(C)). It provides at the same time an algorithm for computing the equisingularity ideal of J. Wahl. The algorithms have been implemented in the computer algebra system Singular. We show them at work by considering two non-trivial examples. As the article is also meant for non-specialists in singularity theory, we include a short survey on new methods and results about equisingularity in characteristic 0.
On Hodge Theory of Singular Plane Curves
Canadian Mathematical Bulletin, 2016
The dimensions of the graded quotients of the cohomology of a plane curve complement U = ℙ2 \ C with respect to the Hodge filtration are described in terms of simple geometrical invariants. The case of curves with ordinary singularities is discussed in detail. We also give a precise numerical estimate for the difference between the Hodge filtration and the pole order filtration on H 2(U,ℂ).
1998
We complete the proof of the fact that the moduli space of rank two bundles with trivial determinant embeds into the linear system of divisors on Picg−1CPic^{g-1}CPicg−1C which are linearly equivalent to 2Theta2\Theta2Theta. The embedded tangent space at a semi-stable non-stable bundle xioplusxi−1\xi\oplus\xi^{-1}xioplusxi−1, where xi\xixi is a degree zero line bundle, is shown to consist of those divisors in ∣2Theta∣|2\Theta|∣2Theta∣ which contain Sing(Thetaxi)Sing(\Theta_{\xi})Sing(Thetaxi) where Thetaxi\Theta_{\xi}Thetaxi is the translate of Theta\ThetaTheta by xi\xixi. We also obtain geometrical results on the structure of this tangent space.
The complex symplectic moduli spaces of uni-modal parametric plane curve singularities
Classification of zero-modal singularities of parametric plane curves under dieomorphism equivalence is extended to uni-modal singularities. Both the simple and uni-modal singulari- ties of parametric plane curves are classified further under symplectomorphic equivalence. In particular the corresponding cyclic symplectic moduli spaces are reconstructed as a canonical ambient spaces for the dieomorphism moduli spaces which are no longer Hausdor spaces.
Polar Invariants of Plane Curves and the
2015
We prove a factorization theorem for the polars of plane singularities with respect to the Newton diagram and calculate the polar quotients of nondegenerated singularities.