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Alon Levy

  1. An Algebraic Proof of Thurston's Rigidity for a Polynomial.

    Ср, 01/11/2012 - 15:01 — SomNambul
    Authors: Alon Levy
    Subjects: Algebraic Geometry
    Abstract

    We study rational self-maps of $\mathbb{P}^{1}$ whose critical points all
    have finite forward orbit. Thurston's rigidity theorem states that outside a
    single well-understood family, there are finitely many such maps over
    $\mathbb{C}$ of fixed degree and critical orbit length. We provide an algebraic
    proof of this fact for polynomial maps, valid over any field whose
    characteristic is zero or larger than the degree of the map. We also produce
    counterexamples when the characteristic of the field is positive and smaller
    than the degree.

  2. The Semistable Reduction Problem for the Space of Morphisms on $\mathbb{P}^{n}$.

    Втр, 04/26/2011 - 15:01 — SomNambul
    Authors: Alon Levy
    Subjects: Algebraic Geometry
    Abstract

    We restate the semistable reduction theorem from geometric invariant theory
    in the context of spaces of morphisms on $\mathbb{P}^{n}$. For every complete
    curve $C$ downstairs, we get a $\mathbb{P}^{n}$-bundle on an abstract curve $D$
    mapping finite-to-one onto $C$, whose trivializations correspond to not
    necessarily complete curves upstairs with morphisms corresponding to
    identifying each fiber with the morphism the point represents.

  3. The Space of Morphisms on Projective Space.

    Втр, 08/25/2009 - 22:41 — SomNambul
    Authors: Alon Levy
    Subjects: Dynamical Systems
    Abstract

    The theory of moduli of morphisms on P^n generalizes the study of rational
    maps on P^1. This paper proves three results about the space of morphisms on
    P^n of degree d > 1, and its quotient by the conjugation action of PGL(n+1).
    First, we prove that this quotient is geometric, and compute the stable and
    semistable completions of the space of morphisms. This strengthens previous
    results of Silverman, as well as of Petsche, Szpiro, and Tepper. Second, we
    bound the size of the stabilizer group in PGL(n+1) of every morphism in terms
    of only n and d.

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