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On the Jacobian Scheme of a plane curve

Stefano Canino, Alessandro Gimigliano, Monica Idà

Abstract

We study the Jacobian scheme of a plane algebraic curve at an ordinary singularity, characterizing it through a geometric property. We compute the Tjurina number for a family of curves at an ordinary singularity showing that it reaches the minimum possible value, using very elementary methods, essentially Gröbner basis. We give an algorithm that gives the analytic type of a double point using the algebraic version of the Mather-Yau Theorem.

On the Jacobian Scheme of a plane curve

Abstract

We study the Jacobian scheme of a plane algebraic curve at an ordinary singularity, characterizing it through a geometric property. We compute the Tjurina number for a family of curves at an ordinary singularity showing that it reaches the minimum possible value, using very elementary methods, essentially Gröbner basis. We give an algorithm that gives the analytic type of a double point using the algebraic version of the Mather-Yau Theorem.
Paper Structure (7 sections, 19 theorems, 69 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 7 sections, 19 theorems, 69 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. The Mather-Yau Theorem for algebraic curves
  4. Jacobian schemes at ordinary singularities
  5. Ordinary singularities with $\tau < \mu$
  6. Jacobian schemes at double points
  7. A remark on the Tjurina number

Key Result

Lemma 2.2

Let $C: f(x_0,x_1,x_2)=0$ be a reduced curve of degree $d$ in ${\fam\msbfam P}^2$ and $P\in C$ with $m_P(C)=m\geq 2$. Then: a) The curve $C$ contains ${\fam\msbfam X}(C)$. b) $m_P(C_i)\geq m-1$ and for at least one of the $C_i$ the multiplicity at $P$ is exactly $m-1$. c) In particular, ${\fam\msbfa

Figures (3)

  • Figure 1: The blue areas give $\ell(Y)$.
  • Figure 2: The domain (in blue) of $\ell_1$.
  • Figure 3: The domain (in blue) of $\ell_4$.

Theorems & Definitions (26)

  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Theorem 3.5
  • Remark 4.1
  • Definition 4.2
  • ...and 16 more