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

TL;DR

This paper establishes new combinatorial bounds on the total Tjurina numbers for specific plane curve arrangements with isolated singularities, and applies these results to construct surfaces with large Tjurina numbers in projective space.

Contribution

It introduces sharp lower bounds for total Tjurina numbers of free line and conic arrangements, and constructs surfaces with arbitrarily large Tjurina numbers, advancing understanding of singularity complexity.

Findings

Total Tjurina number grows quadratically with the number of lines and conics.

Derived sharp structural inequalities for arrangements with ordinary quasi-homogeneous singularities.

Constructed surfaces with arbitrarily large total Tjurina numbers in projective space.

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.