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The projective analytic spectrum of the double of a module

Terence Gaffney, Thiago da Silva

TL;DR

This work analyzes the projectivized analytic spectrum of the double of a module, establishing that the off-diagonal fibers of the Projan of the double decompose as joins of the corresponding fibers at each point. It provides a detailed framework relating $\mathrm{Projan}(\mathcal{R}(M_D))$ to $\mathrm{Projan}(\mathcal{R}(M))$ and its extended/reduced doubles, and demonstrates how these fibers encode limiting tangent data for pairs of points on a variety. Special attention is given to curves on hypersurfaces, where the fiber over the origin can be described in terms of conormal data and the second intrinsic derivative, with explicit plane-curve results illustrating the geometry of limiting tangents and secants.

Abstract

In this work, we investigate the projectivized analytic spectrum of the double of a module, establishing some general properties, and we apply these results to $\mbox{Projan}(\cR((JM(X))_D))$ over the origin in $C\times C$, where $C$ is an irreducible curve in a hypersurface $X$.

The projective analytic spectrum of the double of a module

TL;DR

This work analyzes the projectivized analytic spectrum of the double of a module, establishing that the off-diagonal fibers of the Projan of the double decompose as joins of the corresponding fibers at each point. It provides a detailed framework relating to and its extended/reduced doubles, and demonstrates how these fibers encode limiting tangent data for pairs of points on a variety. Special attention is given to curves on hypersurfaces, where the fiber over the origin can be described in terms of conormal data and the second intrinsic derivative, with explicit plane-curve results illustrating the geometry of limiting tangents and secants.

Abstract

In this work, we investigate the projectivized analytic spectrum of the double of a module, establishing some general properties, and we apply these results to over the origin in , where is an irreducible curve in a hypersurface .

Paper Structure

This paper contains 3 sections, 17 theorems, 34 equations.

Table of Contents

  1. Background on the double of a module
  2. General results about the Projan of the Rees Algebra of the double
  3. The fiber of the Projan over the origin for curves in hypersurfaces

Key Result

Proposition 1.2

Suppose that ${\mathcal{M}}$ is generated by $\{h_1,\ldots,h_r \}$. Then, the following sets are generators of ${\mathcal{M}}_D$:

Theorems & Definitions (36)

  • Definition 1.1
  • Proposition 1.2: SG, Proposition 3.3
  • Proposition 1.3: SG, Proposition 3.5
  • Proposition 1.4: SG, Proposition 3.11
  • Proposition 1.5: SG, Proposition 3.7
  • Definition 1.6
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • Remark 2.3
  • ...and 26 more