The Complete Intersection Discrepancy of a Curve I: Numerical Invariants

1. Problem Statement

The paper addresses the challenge of computing numerical invariants (such as multiplicity, delta invariant, Milnor number, and arithmetic genus) for a curve $X$ that is not necessarily a complete intersection, but is embedded in a larger ambient space.

Specifically, given a Cohen-Macaulay curve $X$ embedded in a smooth variety (or projective space $\mathbb{P}^n_k$), the authors seek to relate the invariants of $X$ to those of a complete intersection curve $Z$ that contains $X$ and agrees with $X$ at the generic points of its irreducible components. The "difference" between $X$ and the residual curve $W = Z \setminus X$ (where $X$ and $W$ are geometrically linked by $Z$) is quantified by a new invariant called the Complete Intersection Discrepancy, denoted $I(X, W)$.

The core problem is to compute $I(X, W)$ without explicitly determining the ideal of the residual curve $W$, which is often computationally difficult.

2. Methodology

The authors employ a combination of Grothendieck duality theory, linkage theory, and local algebraic geometry.

• Adjunction-Type Identity: The central technical tool is an identity derived from Grothendieck duality for finite morphisms. For a Cohen-Macaulay curve $X$ and a Gorenstein complete intersection $Z$ containing $X$, they establish:
  $$\omega_X \cong i^*(I_W \omega_Z)$$
  where $\omega_X$ and $\omega_Z$ are the dualizing sheaves, $i: X \hookrightarrow Z$ is the inclusion, and $I_W$ is the ideal sheaf of the residual curve $W$ in $Z$.
• Local Analysis of Singularities: The authors analyze local invariants at a point $x \in X$, including:
  • The delta invariant ($\delta_x$).
  • The Milnor number ($\mu_x$).
  • The multiplicity ($m_x$).
  • The ramification ideal ($R_X$) and its Hilbert-Samuel multiplicity $e(R_X)$.
  • The Jacobian ideals of $X$ and $Z$.
• Construction of $Z$: Using prime avoidance and linear algebra, the authors provide an explicit algorithm to construct the complete intersection $Z$ defined by general linear combinations of the equations defining $X$. This ensures $Z$ is reduced along $X$ and shares the same generic points.
• Nash Blowup Interpretation: For Gorenstein curves, the discrepancy is interpreted geometrically via the degree of the exceptional divisor of the Nash blowup.

3. Key Contributions and Results

A. Definition of Complete Intersection Discrepancy

The authors define the Complete Intersection Discrepancy at a point $x$ as:
$$I_x(X, W) := \dim_k \mathcal{O}_{X,x} / (I_X + I_W)$$
The global discrepancy is $I(X, W) = \sum_{x \in X \cap W} I_x(X, W)$.
Crucially, they prove that for a general choice of $Z$, this value is an intrinsic invariant of the singularity of $X$, denoted $\text{cid}(X, x)$, independent of the specific choice of $Z$.

B. Generalized Multiplicity Formula (Theorem A)

The paper generalizes classical formulas (due to Lê, Greuel, and Teissier) for plane and complete intersection curves. For a general complete intersection $Z$ containing $X$:
$$e(\text{Jac}(X, x)) - \text{cid}(X, x) = 2\delta_x + e(R_X)$$
If $x$ is a tame point (e.g., characteristic 0), this simplifies to:
$$e(\text{Jac}(X, x)) - \text{cid}(X, x) = \mu_x + m_x - 1$$
This relates the Hilbert-Samuel multiplicity of the Jacobian ideal to the Milnor number and the ramification of the normalization.

C. Generalized Genus-Degree Formula (Theorem B)

The authors derive a formula for the arithmetic genus $p_a(X)$ of a Cohen-Macaulay curve $X \subset \mathbb{P}^n_k$ that is generically a complete intersection. If $Z$ is defined by polynomials of degrees $d_1, \dots, d_{n-1}$:
$$p_a(X) = 1 + \frac{(d_1 + \dots + d_{n-1} - n - 1)\deg(X) - I(X, W)}{2}$$
This generalizes the classical genus-degree formula for smooth complete intersections.

• Computational Advantage: When $X$ is smooth, $I(X, W)$ can be computed directly from the Jacobian ideal of $Z$ without finding the ideal of $W$:
  $$I(X, W) = \dim_k S(X)(h) / \text{Jac}(Z)$$
  where $S(X)(h)$ is the degree-zero part of the localization of the coordinate ring at a general hyperplane $h$.

D. Almost Complete Intersection (ACI) Curves (Corollary C)

For an ACI curve (defined by $n$ equations in $\mathbb{P}^n$), the discrepancy has a closed form:
$$I(X, W) = \prod_{i=1}^n d_i - d_n \deg(X)$$
This leads to a lower bound on the degree of non-degenerate ACI curves.

E. Geometric Interpretation via Nash Blowup

For a Gorenstein curve $X$, the quantity $e(\text{Jac}(X, x)) - \text{cid}(X, x)$ is shown to be the degree of the exceptional divisor of the Nash blowup of $X$ at $x$. This answers a question posed by Piene regarding the center of the Nash blowup for Gorenstein schemes.

F. Transversality and Computability

The paper demonstrates that for general choices of equations, the intersection $X \cap W$ consists of ordinary double points of $Z$. Furthermore, the authors provide an algorithmic approach (Section 6.4) to compute $I(X, W)$ using computer algebra systems (like Singular) by calculating vector space dimensions of specific ideals, avoiding the need for primary decomposition.

4. Significance

• Unification of Invariants: The paper unifies various classical results in singularity theory (Lê-Greuel-Teissier formulas) and algebraic geometry (genus-degree formulas) under a single framework involving the complete intersection discrepancy.
• Computational Efficiency: It provides a practical method to compute the arithmetic genus and other invariants for singular curves that are not complete intersections, bypassing the difficult task of computing the ideal of the residual curve $W$.
• Equisingularity: The discrepancy serves as a tool for studying equisingularity in families of curves, as it remains constant under flat deformations (Corollary 5.4).
• Generalization: The results extend beyond plane curves to curves of arbitrary codimension in projective space, including those with locally smoothable singularities.
• Theoretical Depth: The use of Grothendieck duality to derive an adjunction formula for non-Gorenstein schemes embedded in Gorenstein schemes provides a robust theoretical foundation for linkage theory in higher dimensions.

In summary, the paper introduces a robust numerical invariant ($I(X, W)$) that bridges the gap between the geometry of a curve and its complete intersection envelope, offering both deep theoretical insights and practical computational tools for algebraic geometers.