Plane curve singularities and Fitting ideals

Abstract: In this note we investigate the Fitting ideals associated to the Tjurina ideal of a non quasi-homogeneous plane curve singularity. Special properties occur when the difference between Milnor number and Tjurina number is at most 2.

• The paper demonstrates that syzygy modules and Fitting ideals characterize invariants of non quasi-homogeneous plane curve singularities.
• It provides a minimal free resolution of the Tjurina ideal and links Milnor-Tjurina differences with the codimension of specific Fitting ideals.
• Explicit computational examples verify theoretical claims and illustrate the complex behavior of algebraic invariants in higher-codimension cases.

Fitting Ideals Associated to Non-Quasi-Homogeneous Plane Curve Singularities

Introduction and Context

The paper "Plane curve singularities and Fitting ideals" (2606.20343) investigates algebraic invariants arising from the structure of plane curve singularities, focusing particularly on the Fitting ideals associated with the Tjurina ideal for non quasi-homogeneous cases. The study is set in the context of the convergent power series ring $R = \mathbb{C}\{x, y\}$, considering an isolated singularity $X: f = 0$ at the origin. Central objects are the Jacobian ideal $J_f = (f_x, f_y)$, the Tjurina ideal $I_f = (f_x, f_y, f)$, and their corresponding Artinian algebras $M(f)$ and $T(f)$ with dimensions $\mu(f)$ (Milnor number) and $\tau(f)$ (Tjurina number), respectively.

The singularity is quasi-homogeneous precisely when $I_f = J_f$, equivalently $\mu(f) = \tau(f)$. This regime has well-understood resolutions and syzygies. The paper restricts attention to the more intricate non quasi-homogeneous case, where the syzygy module $X: f = 0$0 and the module of logarithmic derivations $X: f = 0$1 play crucial roles in the analysis of minimal resolutions and Fitting ideals.

Structure and Properties of Syzygies and Fitting Ideals

The paper establishes that $X: f = 0$2 is a free $X: f = 0$3-module of rank 2, associated with pairs of syzygies $X: f = 0$4, $X: f = 0$5. This allows a minimal free resolution of $X: f = 0$6:

$X: f = 0$7

where $X: f = 0$8 and $X: f = 0$9.

The Fitting ideals $J_f = (f_x, f_y)$0 are computed as the ideals of $J_f = (f_x, f_y)$1-minors of the matrix $J_f = (f_x, f_y)$2 associated to $J_f = (f_x, f_y)$3, with the key identifications:

• $J_f = (f_x, f_y)$4
• $J_f = (f_x, f_y)$5

An important structural result is that $J_f = (f_x, f_y)$6 is minimally generated by two elements, yielding a zero-dimensional complete intersection.

Milnor–Tjurina Difference and Fitting Ideals

The central focus is the relationship between the Milnor number $J_f = (f_x, f_y)$7, the Tjurina number $J_f = (f_x, f_y)$8, and the codimension of the Fitting ideal $J_f = (f_x, f_y)$9. The results are:

• $I_f = (f_x, f_y, f)$0
• $I_f = (f_x, f_y, f)$1, with equality in cases where $I_f = (f_x, f_y, f)$2.

A strong claim is established for the case $I_f = (f_x, f_y, f)$3: $I_f = (f_x, f_y, f)$4 and $I_f = (f_x, f_y, f)$5. The paper provides explicit classification data for $I_f = (f_x, f_y, f)$6 in terms of isolated complete intersections, characterizing their generators in terms of monomials $I_f = (f_x, f_y, f)$7 for $I_f = (f_x, f_y, f)$8, with higher-codimension cases exhibiting more subtle behavior.

Examples and Computational Evidence

The authors provide explicit computations (using SINGULAR) verifying the theoretical claims for concrete curve singularities. For instance:

• For $I_f = (f_x, f_y, f)$9, $M(f)$0, $M(f)$1, and $M(f)$2.
• For $M(f)$3, $M(f)$4, $M(f)$5, and $M(f)$6.
• Contradictory cases arise for $M(f)$7, where $M(f)$8, illustrating failure of the equality in higher codimension.

Numerical results highlight that the difference $M(f)$9 can be significant for certain singularities, e.g., up to 14.

Implications and Theoretical Significance

The structural results concerning Fitting ideals offer new insights into the fine algebraic geometry of non quasi-homogeneous plane curve singularities. When $T(f)$0, the Fitting ideal $T(f)$1 tightly controls the defect of quasi-homogeneity. In higher-codimension cases, the behavior is more complex and cannot be encapsulated by simple monomial generators, suggesting deeper intricacies in the corresponding singularity moduli.

Practical implications include the identification of invariants for classifying singularities and their moduli, and the efficient explicit computation of algebraic invariants using computer algebra systems. Theoretically, the analysis bridges syzygies, free resolutions, and Fitting ideals, providing a robust framework for understanding the local algebraic structure of singularities beyond the quasi-homogeneous regime.

Future Directions

The results suggest avenues for further research:

• Extension to isolated surface singularities, where the structure of syzygies and Fitting ideals is less tractable, as exemplified by explicit computations for $T(f)$2.
• Investigation of cases where $T(f)$3, for which classification and equality conditions are more subtle.
• Deepening the study of deformation theory and equisingularity invariants via Fitting ideals of higher-order syzygy modules.

Conclusion

The paper rigorously characterizes the interplay between the Milnor and Tjurina numbers of plane curve singularities and the structure of associated Fitting ideals, notably elucidating cases where $T(f)$4. The theoretical and computational framework developed enhances the understanding of non quasi-homogeneous singularities, both for their algebraic classification and for potential algorithmic applications in singularity theory.