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Ciliberto-Di Gennaro conjecture for sextic hypersurfaces

Ksenia Kvitko

TL;DR

The paper proves the Ciliberto–Di Gennaro conjecture for sextic nodal hypersurfaces in P^4 (degree d=6), showing that such a threefold is factorial unless it contains a plane or a quadric with prescribed node counts. It adapts Kloosterman’s technique, employing Hilbert-function bounds and defect arguments via artinian Gorenstein rings to constrain the singular locus and detect complete intersections inside it. The main result follows by ruling out potential exceptional h-vector configurations and deriving that non-factoriality would force geometric configurations (plane or quadric) that align with the conjecture's stated alternatives. This advances factoriality results for nodal threefolds and informs broader questions in birational geometry and rationality.

Abstract

The Ciliberto-Di Gennaro conjecture addresses the factoriality of three-dimensional nodal hypersurfaces, and their geometric properties. We prove this conjecture for hypersurfaces of degree 6 by adapting a recent technique due to R. Kloosterman.

Ciliberto-Di Gennaro conjecture for sextic hypersurfaces

TL;DR

The paper proves the Ciliberto–Di Gennaro conjecture for sextic nodal hypersurfaces in P^4 (degree d=6), showing that such a threefold is factorial unless it contains a plane or a quadric with prescribed node counts. It adapts Kloosterman’s technique, employing Hilbert-function bounds and defect arguments via artinian Gorenstein rings to constrain the singular locus and detect complete intersections inside it. The main result follows by ruling out potential exceptional h-vector configurations and deriving that non-factoriality would force geometric configurations (plane or quadric) that align with the conjecture's stated alternatives. This advances factoriality results for nodal threefolds and informs broader questions in birational geometry and rationality.

Abstract

The Ciliberto-Di Gennaro conjecture addresses the factoriality of three-dimensional nodal hypersurfaces, and their geometric properties. We prove this conjecture for hypersurfaces of degree 6 by adapting a recent technique due to R. Kloosterman.
Paper Structure (5 sections, 22 theorems, 82 equations, 1 table)

This paper contains 5 sections, 22 theorems, 82 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structure of artinian Gorenstein rings
  4. Estimates for values of Hilbert function
  5. Nodal hypersurfaces of degree 6

Key Result

Theorem 1.2

Conjecture conj holds for $d=6$.

Theorems & Definitions (48)

  • Conjecture 1.1
  • Theorem 1.2
  • Definition 2.1
  • Theorem 2.2: Macaulay macaulay1927some
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6
  • ...and 38 more