Total Tjurina Number in Singular Curves

• Total Tjurina number is a global analytic invariant that summarizes the complexity of local singularities in algebraic plane curves.
• It integrates analytic, valuation, and syzygy data to govern deformation theory, homological properties, and combinatorial classifications.
• Extremal values of the total Tjurina number classify free, nearly free, and minimal-Tjurina curves, guiding the stratification of singularity types.

The total Tjurina number is a central analytic invariant in the study of singularities of algebraic hypersurfaces, particularly complex plane curves. Defined globally as the sum of local Tjurina numbers at the singular points, it controls key deformation-theoretic, homological, and combinatorial properties of the curve. Its computation integrates local analytic, valuation-theoretic, and syzygy-theoretic information, and its extremal values classify important classes such as free, nearly free, and minimal-Tjurina curves.

1. Definition and Fundamental Properties

Given a reduced complex projective plane curve $C : f(x,y,z) = 0$ of degree $d$, the local Tjurina number at an isolated singularity $p \in \operatorname{Sing} C$ is defined as

$$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$

The total (global) Tjurina number is then

$$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$

Equivalently, $\tau(C)$ is the colength of the Jacobian ideal $\langle f_x, f_y, f_z \rangle$ in the homogeneous coordinate ring; it coincides with the degree of the singularity subscheme defined by the Jacobian ideal or with $\dim_{\mathbb{C}} M(f)_{k \gg 0}$, where $M(f) = S / J_f$ and $S = \mathbb{C}[x,y,z]$ (Dimca, 2015, Dimca et al., 2019, Dimca et al., 13 Jan 2026).

For local isolated hypersurface singularities $d$0, the local Tjurina number

$d$1

coincides with $d$2 (Milnor number) precisely for quasihomogeneous singularities, and $d$3 in general (Hu et al., 2024).

The total Tjurina number is semicontinuous in families and provides the dimension of the tangent space to the base of the miniversal deformation.

2. Homological Interpretation and Syzygies

The computation and extremal values of $d$4 are intimately related to the syzygies among the Jacobian partials.

Let $d$5 denote the module of Jacobian syzygies. Let $d$6 be the minimal degree of a nontrivial syzygy. Then,

• Free curves: $d$7, $d$8, and $d$9 with $p \in \operatorname{Sing} C$0.
• Nearly free curves: $p \in \operatorname{Sing} C$1 minimally generated by three syzygies of degrees $p \in \operatorname{Sing} C$2, $p \in \operatorname{Sing} C$3, and $p \in \operatorname{Sing} C$4.

The minimal graded free resolution of $p \in \operatorname{Sing} C$5 encodes the graded Betti numbers $p \in \operatorname{Sing} C$6, leading to a closed formula: $p \in \operatorname{Sing} C$7 The sequence of exponents $p \in \operatorname{Sing} C$8 and shifts $p \in \operatorname{Sing} C$9 associated to this resolution stratify the moduli of singular curves according to the structure of their syzygy modules (Dimca et al., 13 Jan 2026, Dimca et al., 2018).

3. Extremal and Inequality Results

A key structural result is the du Plessis–Wall bound: $$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$0 If $$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$1, the upper bound is attained if and only if $$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$2 is free. For $$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$3, nearly free curves attain the maximal possible value, $$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$4. For $$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$5, a refined upper bound applies: $$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$6 Equality cases characterize special classes:

• $$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$7: free curves
• $$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$8: nearly free
• $$\tau_p(C) = \dim_\mathbb{C} \left( \mathcal{O}_{\mathbb{P}^2, p} / (f, \partial_x f, \partial_y f, \partial_z f)_p \right).$$9: minimal Tjurina curves (e.g., certain Thom–Sebastiani curves and union of a smooth curve with a transverse line) (Dimca, 2015, Dimca et al., 2019, Ellia, 2019, Dimca et al., 2018).

For reducible plane curves, the framework generalizes to

$$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$0

where $$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$1 is the total intersection multiplicity between partitions, with sharpness in quasi-homogeneous cases (Hefez et al., 2024).

4. Local and Valuation-Theoretic Aspects

Locally, for a reduced curve singularity $$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$2, the Tjurina number may also be interpreted as the length (colength) of the Jacobian ideal in the local ring, and has intimate connections to value semigroups, Puiseux characteristics, and Kähler differentials.

For irreducible branches, Berger–Greuel gives $$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$3, $$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$4 the Kähler value semigroup, $$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$5 the value semigroup of the branch (Almirón et al., 22 Jan 2025, Hernandes et al., 2019).

For plane curves with $$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$6 branches, Hefez–Hernandes provide an exact additive formula: $$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$7 where the last term captures purely combinatorial differences of value-sets of fractional ideals. This approach both refines and proves inequalities conjectured by Dimca (Hefez et al., 2024).

In cases of two branches with high intersection multiplicity, the formula $$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$8 holds, where $$\tau(C) = \sum_{p \in \operatorname{Sing} C} \tau_p(C).$$9 is the intersection multiplicity and $\tau(C)$0 the conductor of the common value semigroup (Almirón et al., 22 Jan 2025).

5. Weighted Homogeneous and Higher-Order Tjurina Numbers

For isolated weighted homogeneous hypersurface singularities, explicit formulas for the total Tjurina number are available: $\tau(C)$1 where $\tau(C)$2 are the weights of $\tau(C)$3. The $\tau(C)$4th Tjurina numbers $\tau(C)$5, defined as $\tau(C)$6, measure higher-order obstructions to deformations. For $\tau(C)$7 below the minimal multiplicity of partials, $\tau(C)$8 admits a purely combinatorial formula, but for $\tau(C)$9 large, the dependence is refined by the structure of syzygies and the detailed Koszul complex (Hu et al., 2024).

6. Geometric, Topological, and Deformation-Theoretic Roles

The total Tjurina number governs the base space dimension for the miniversal deformation of a hypersurface singularity—counting directions for analytic (not merely topological) smoothing. It provides finer invariants than the Milnor number, especially for non-quasihomogeneous singularities, and captures the codimension of the equisingularity stratum in the space of all hypersurface singularities.

In the context of foliations, the Tjurina number of the foliation (for a balanced divisor of separatrices) relates via the Gómez–Mont–Seade–Verjovsky index to that of the underlying curve, adding further structure to the representation theory of singularities and yielding global bounds on the sum of Milnor numbers on projective spaces (Fernández-Pérez et al., 2021).

7. Applications and Open Problems

Applications encompass:

• Classification of free, nearly free, and minimal Tjurina plane curves (Dimca, 2015, Dimca et al., 2018).
• Stratification of modular families of curve singularities according to Betti patterns (Dimca et al., 13 Jan 2026).
• Explicit numerical invariants for families of curves with fixed topological or analytic data, particularly for curves with many branches or high intersection multiplicity (Almirón et al., 22 Jan 2025, Hefez et al., 2024).
• Computation of $\langle f_x, f_y, f_z \rangle$0 for large classes, such as line arrangements and Brieskorn–Pham curves (Dimca et al., 2019, Hu et al., 2024).

Open questions include full numerical characterizations for reducible singularities beyond the quasi-homogeneous case, stratification of Betti-number patterns, and extension of these results to higher dimensions and non-isolated singularities (Dimca et al., 13 Jan 2026, Hefez et al., 2024).