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Maximal families of nodal varieties with defect

Remke Kloosterman

Abstract

In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large.

Maximal families of nodal varieties with defect

Abstract

In this paper we prove that a nodal hypersurface in P^4 with defect has at least (d-1)^2 nodes, and if it has at most 2(d-2)(d-1) nodes and d>6 then it contains either a plane or a quadric surface. Furthermore, we prove that a nodal double cover of P^3 ramified along a surface of degree 2d with defect has at least d(2d-1) nodes. We construct the largest dimensional family of nodal degree d hypersurfaces in P^(2n+2) with defect for d sufficiently large.

Paper Structure

This paper contains 5 sections, 20 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Macaulay's and Green's result
  3. Nodal complete intersections
  4. Hypersurfaces with defect
  5. The Ciliberto-Di Gennaro conjecture

Key Result

Theorem 1.1

Let $X\subset \mathbf{P}^4$ be a nodal hypersurface of degree at least 3. Assume that $h^4(X)\geq 2$. Then $X$ has at least $(d-1)^2$ nodes. Moreover, if equality holds then $X$ contains a plane.

Theorems & Definitions (38)

  • Theorem 1.1: Cheltsov ChelFacChelPlane
  • Theorem 1.2
  • Theorem 1.3: Cheltsov Chelwps
  • Theorem 1.4
  • Theorem 2.1: Macaulay Mac
  • Corollary 2.2
  • Corollary 2.3
  • Theorem 2.4: Gotzmann Gotz
  • Corollary 2.5
  • proof
  • ...and 28 more