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 $\mathbb{P}<sup>2$ 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 $\mathbb{P}<sup>{3}$ 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.