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On combinatorial bounds for the total Tjurina numbers of certain curves and surfaces with isolated singularities

Piotr Pokora

Abstract

We investigate combinatorial bounds for the total Tjurina numbers of plane curve arrangements. Focusing on arrangements of lines and conics in $\mathbb{P}^2$ that admit only ordinary quasi-homogeneous singularities, we derive new structural inequalities governing the distribution of multiple intersection points. As a consequence, we establish sharp lower bounds for the total Tjurina numbers of free line arrangements with bounded maximal multiplicity and, more generally, for free conic-line arrangements. In particular, we show that for a free arrangement of $d$ lines and $k$ conics, the total Tjurina number grows at least quadratically in $d$ and $k$, and we demonstrate that this bound is sharp. As an application of these planar results, we construct a family of surfaces in $\mathbb{P}^{3}$ with only isolated singularities and arbitrarily large total Tjurina numbers.This provides new lower bounds for the total Tjurina numbers of certain hypersurfaces that are independent of detailed homological data.

On combinatorial bounds for the total Tjurina numbers of certain curves and surfaces with isolated singularities

Abstract

We investigate combinatorial bounds for the total Tjurina numbers of plane curve arrangements. Focusing on arrangements of lines and conics in that admit only ordinary quasi-homogeneous singularities, we derive new structural inequalities governing the distribution of multiple intersection points. As a consequence, we establish sharp lower bounds for the total Tjurina numbers of free line arrangements with bounded maximal multiplicity and, more generally, for free conic-line arrangements. In particular, we show that for a free arrangement of lines and conics, the total Tjurina number grows at least quadratically in and , and we demonstrate that this bound is sharp. As an application of these planar results, we construct a family of surfaces in with only isolated singularities and arbitrarily large total Tjurina numbers.This provides new lower bounds for the total Tjurina numbers of certain hypersurfaces that are independent of detailed homological data.
Paper Structure (5 sections, 11 theorems, 52 equations)

This paper contains 5 sections, 11 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Results concerning complex line arrangements with small multiplicities
  4. A lower bound on the total Tjurina numbers of some conic-line arrangements
  5. An application towards surfaces with only isolated singularities

Key Result

Theorem 2.3

Let $C = \{f=0\}\subset \mathbb{P}^{2}$ be a reduced curve of degree $m$ and $d_1 = {\rm mdr}(f)$. Let us denote the total Tjurina number of $C$ by $\tau(C)$. Then the following two cases hold.

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: du Plessis -- Wall, DPP1
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • ...and 13 more