The Complete Intersection Discrepancy of a Curve I: Numerical Invariants
Andrei Benguş-Lasnier, Antoni Rangachev
TL;DR
The paper introduces complete intersection discrepancy $I(X,W)$ as a local correction term that enables linking invariants of a Cohen–Macaulay curve $X$ to a surrounding complete intersection $Z$ via an adjunction-type identity $\omega_X = i^*(\mathcal{I}_W \omega_Z)$ derived from Grothendieck duality. It then generalizes key classical formulas, notably the Lè-Teissier multiplicity relations and the genus–degree formula, by expressing local and global invariants through $I(X,W)$ and related ramification data, yielding the fundamental equality $e( Jac(Z,x)) - I_x(X,W) = 2\delta_x + e(R_X)$. For Gorenstein curves, $I(X,W)$ ties to the Nash blowup center, and in the smooth case the arithmetic genus satisfies $p_a(X) = 1 + \frac{\deg(X)(d_1+\cdots+d_{n-1}-n-1) - I(X,W)}{2}$. The work also provides constructive methods to build a suitable CI $Z$, computes $I(X,W)$ in broad cases (including smooth, locally complete intersection, and smoothable $X$), and develops transversality and computability results to support practical calculations and deformations of curves. These advances offer a robust framework for understanding equisingularity, linking, and genus corrections across families of curves.
Abstract
We generalize two classical formulas for complete intersection curves using the complete intersection discrepancy of a curve as a correction term. The first formula is a well-known multiplicity formula in singularity theory due to Lê, Greuel and Teissier that relates some of the basic invariants of a curve singularity. We apply its generalization elsewhere to the study of equisingularity of curves. The second formula is the genus--degree formula for projective curves. The main technical tool used to obtain these generalizations is an adjunction-type identity derived from Grothendieck duality theory.