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3 papers · Page 1 of 1

Algebra

Edition 1

Concavity of Lattice Counts on Toric Threefolds

Scott Schubr

Primary subject
Mathematics
Subject subfields
Algebra · Geometry
Paper type
Research Article
Manuscript stage
Preprint

The fourth root of the ordinary Riemann-Roch count polynomial is concave on the entire closed nef cone of every smooth toric Fano threefold. This exponent is uniformly optimal for the ordinary real polynomial extensions. Precisely nine of the eighteen threefolds have concave cube root; the other nine have cube-root failure, certified by exact fan and polynomial computations. Integral nonbig nef classes are included. For split projective-line bundles over the projective plane, the ordinary polynomial has optimal power one third at twists zero, one and two, while no positive concave power exists at any integer twist at least three. Geometrically weighted lattice counts with bias at least one admit nonpolynomial concave logarithmic extensions at every twist, whereas every fixed bias strictly between zero and one has integral midpoint counterexamples at large twists. Ordinary-polynomial optimality and lattice-only sharpness are distinguished.

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    Complex Analysis

    Edition 1

    Compact capture and protected bridges on the Koras–Russell cylinder

    Scott Schubr

    Primary subject
    Mathematics
    Subject subfields
    Complex Analysis · Geometry
    Paper type
    Research Article
    Manuscript stage
    Preprint

    On an adapted model of the Koras-Russell cylinder, we give explicit compression of a closed C-star carrier and a sufficient criterion for ambient compact capture: the planar polynomial hull of the scalar image under q = xy+1 must avoid 1. This yields fixed Runge capture domains, including the connected inverse image of the plane slit along [1,infinity). Joint capture can be corrected to approximate the identity on a globally holomorphically convex parametrized ball inside a fixed Runge Euclidean domain. Prescribed compact bridges of positive thickness can thus be captured while protecting the ball without scalar-range separation. A complementary convex-ball argument treats certain protected bodies through q=1, and an analytic disk explains the obstruction to extending weighted contraction to the entire punctured chart. These are qualitative capture criteria, not a biholomorphic identification of the whole cylinder with C^4 or an improved stabilization bound.

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      Edition 2

      Finite-projection stabilization of contractible affine hypersurfaces

      Scott Schubr

      Primary subject
      Mathematics
      Subject subfields
      Complex Analysis · Geometry
      Paper type
      Research Article
      Manuscript stage
      Preprint

      Let X be a smooth contractible affine hypersurface of complex dimension n >= 2 with the holomorphic density property. We prove that X times C^n is biholomorphic to C^(2n). A finite monic projection replaces the proper-embedding parameter in Huang’s stabilization construction, retaining colliding fibers as curvilinear finite schemes. Stratified Oka interpolation and parametric Andersén-Lempert approximation yield compact capture in a fixed Runge Euclidean basin. Applications include McLean’s varieties and the Koras-Russell cubic. The sufficient ambient dimensions in specified nonlinearizable-action constructions are lowered by one, including disconnected cyclic fixed loci in C^6 and a fixed contractible non-Euclidean surface in C^8.

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