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Bounding Projective Hypersurface Singularities

B. Castor

Abstract

We compare several different methods involving Hodge-theoretic spectra of singularities which produce constraints on the number and type of isolated singularities on projective hypersurfaces of fixed degree. In particular, we introduce a method based on the spectrum of the nonisolated singularity at the origin of the affine cone on such a hypersurface, and relate the resulting explicit formula to Varchenko's bound.

Bounding Projective Hypersurface Singularities

Abstract

We compare several different methods involving Hodge-theoretic spectra of singularities which produce constraints on the number and type of isolated singularities on projective hypersurfaces of fixed degree. In particular, we introduce a method based on the spectrum of the nonisolated singularity at the origin of the affine cone on such a hypersurface, and relate the resulting explicit formula to Varchenko's bound.

Paper Structure

This paper contains 8 sections, 35 theorems, 93 equations.

Table of Contents

  1. A Formula for Hodge Numbers of Smooth Projective Hypersurfaces
  2. The Vanishing Cycle Sequence Method
  3. Varchenko's Bound
  4. Conical Bounding Method for Pham-Brieskorn
  5. Eigenvalue Bounding Method
  6. A generalization of the conical bound
  7. Some more user-friendly formulas
  8. Appendix: comparing the bounds

Key Result

Theorem 1.1

For a smooth hypersurface $X \subseteq \mathbb{P}^{n}$ of degree $d$, the middle primitive Hodge numbers $[h^{k,n-1-k}_{n,d}]'$ (where $k \leq \frac{n-1}{2}$) are given by: In particular, if $d>n$, then:

Theorems & Definitions (64)

  • Theorem 1.1
  • Lemma 1.2
  • proof
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 54 more