Local Complete Intersection Curve

• Local complete intersection curves are subschemes defined by ideals of codimension 2, generated by a regular sequence of length 2, ensuring local regularity.
• They are set-theoretically complete intersections where properties like the triviality of the conormal bundle often imply an ideal-theoretic complete intersection.
• Applications include characterizing monomial curves and moduli spaces of Gorenstein curves, linking affine and projective intersection theory.

A local complete intersection (lci) curve is a central notion in algebraic geometry and commutative algebra, arising in the paper of schemes and their subschemes defined by regular sequences. Specifically, a closed subscheme $C \subset \operatorname{Spec}A$ of a commutative Noetherian ring $A$ is a local complete intersection curve if its ideal %%%%2%%%% has codimension $2$ (i.e., $\operatorname{ht} I = 2$) and, for every prime $\mathfrak{p} \supset I$, the localization $I_{\mathfrak{p}} \subset A_{\mathfrak{p}}$ is generated by a regular sequence of length $2$. This property captures the regularity and the local generation of $C$ as an intersection, and is robust under various operations, making it a key ingredient in both local and global intersection theory.

1. Formal Definition and Invariants

Let $A$ be a commutative Noetherian ring of Krull dimension $3$. A closed subscheme $C \subset \operatorname{Spec}A$ is a local complete intersection (lci) curve if its defining ideal $I \subset A$ satisfies:

• $\operatorname{ht} I = 2$;
• for every $\mathfrak{p} \supset I$, $I_{\mathfrak{p}} \subset A_{\mathfrak{p}}$ is generated by a regular sequence of length $2$.

Equivalently, $I$ is a local complete intersection ideal of height $2$.

The conormal sheaf is a primary invariant: $\mathcal{N}^*_{C/A} := \mathcal{I}/\mathcal{I}^2$, which is a locally free $\mathcal{O}_C$-module of rank $2$ when $I$ is lci. Triviality of the conormal bundle, i.e., $I/I^2 \simeq (A/I)^2$, plays an essential role in characterizing when an lci curve is also a complete intersection in the ideal-theoretic sense (Mandal et al., 10 Nov 2025).

2. Set-Theoretic vs. Ideal-Theoretic Complete Intersections

A distinction is made between:

• Ideal-theoretic complete intersection (ci): An ideal $I \subset A$ of height $r$ is a ci if there exists a regular sequence $f_1, \dots, f_r \in I$ such that $I = (f_1, \dots, f_r)$.
• Set-theoretic complete intersection (stci): $I$ is an stci if there exists a regular sequence $f_1, \dots, f_r \in A$ such that $\sqrt{I} = \sqrt{(f_1, \dots, f_r)}$.

The stci condition is weaker, as it only requires the radical of $I$ to agree with that of a regular-sequence ideal. For lci curves in $\operatorname{Spec}A$, dimension $3$, every lci curve is a set-theoretic complete intersection (Mandal et al., 10 Nov 2025).

Property	Ideal-theoretic CI	Set-theoretic CI
Generators	Regular sequence, $I=(f_1,...,f_r)$	Regular sequence, $\sqrt{I} = \sqrt{(f_1,...,f_r)}$
Implication	CI $\implies$ STCI	Not all STCI are CI

3. Main Results for Local Complete Intersection Curves

Set-Theoretic Generation

The main theorem [(Mandal et al., 10 Nov 2025), Thm 2.2] asserts: Let $A$ be a Noetherian ring of dimension $3$ and $I \subset A$ a local complete intersection ideal of height $2$. Then $I$ is a set-theoretic complete intersection; i.e., there exist $f,g \in A$ (a regular sequence) with $\sqrt{I} = \sqrt{(f,g)}$.

Key Ingredients in the Proof

• Ferrand–Szpiro reduction: Any lci ideal $I$ with $\dim(A/I)\leq 1$ can be replaced by an lci ideal $J$ with $\sqrt{J} = \sqrt{I}$ and $J/J^2$ locally free of rank $2$.
• Serre’s splitting principle: Provides an exact sequence with projective modules, leading to a Koszul-type complex whose exactness is ensured by local properties.
• Generation criterion: If $J$ of height $2$ can be generated by $2$ elements, it is an ideal-theoretic ci.
• Conormal bundle triviality: If $I/I^2$ is a free $A/I$-module of rank $2$, then $I$ is generated by a regular sequence.

Curves with Trivial Conormal Bundle

When $I/I^2 \simeq (A/I)^2$, the ideal $I$ is an ideal-theoretic ci. This is formalized in [(Mandal et al., 10 Nov 2025), Thm 2.4]: If $I$ is lci of height $2$ and $I/I^2$ is free of rank $2$, then $I$ is generated by a regular sequence.

Examples and Applications

• Every smooth space curve in $\mathbb{A}^3$ (over a field) is set-theoretically a complete intersection of two surfaces.
• Twisted cubic in $\mathbb{P}^3$ (affine cone in $\mathbb{A}^4$): its defining ideal is lci, hence set-theoretic ci in the cone.
• For surfaces in affine $4$-space over $\overline{\mathbb{F}}_p$, the analogous results hold, using the Bloch–Murthy–Szpiro strategy.

4. Relation to Global and Singular Settings

Primitive Complete Intersection Structures

For curves in $\mathbb{P}^3$, the concept of a primitive structure (locally contained in a smooth surface) refines the stci condition. For such a curve $C \subset \mathbb{P}^3$, a primitive multiple structure $X$ is defined scheme-theoretically by $X = F_a \cap F_b$, with $F_a, F_b$ surfaces of degrees $a \leq b$, and local analytic equations of the form $(u, v^m)$ along $C$. Numerical conditions involving the genus, degree, and type of the conormal bundle, as well as Miyaoka-type singularity inequalities, govern the possibility of such structures (Ellia, 2014).

• Key identities for existence:
  • $2g - 2 + \ell(m-1) = d(a + b - 4)$,
  • $md = ab$,
  • $\ell \leq 3d + 2g - 4$,
  • Miyaoka-type bounds on singularities.

This framework determines for which curves $C$ there exists a primitive complete-intersection structure making $C$ an stci.

5. Local Complete Intersection Property for Monomial Curves

For monomial curves $C_S$ associated to a numerical semigroup $S \subset \mathbb{N}$, the local complete intersection property is characterized combinatorially. The ideal $I(C_S)$ in $k[X_{a_1},\ldots,X_{a_r}]$ is generated by $r-1$ binomial relations forming a regular sequence if and only if $S$ is a complete-intersection semigroup (Contiero et al., 2022). In this case, the corresponding projective moduli space of Gorenstein curves with symmetric Weierstrass semigroup $S$ is a weighted projective space.

Object	LCI Characterization
Monomial curve $C_S$	$I(C_S)$ generated by $r-1$-element regular sequence
Gorenstein curve	Moduli is (weighted) projective space iff lci holds

6. Dimension Theory and Counterexamples

While in dimension one, every formal complete intersection local integral domain is an absolute complete intersection (i.e., a quotient of a regular local ring by a regular sequence), this fails in higher dimensions (Heitmann et al., 2011). Explicitly, there exists a $3$-dimensional domain whose completion is a complete intersection in the formal sense, but which is not a homomorphic image of a regular local ring by a regular-sequence ideal. Thus, the equivalence of formal and absolute ci for lci rings is restricted to essentially dimension one settings; for curves, this distinction aligns with the classical and formal approaches to lci structures.

A plausible implication is that, for lci curves in dimension three, the set-theoretic generation attained via stci need not always lift to a globally ideal-theoretic complete intersection unless auxiliary conditions (such as trivial conormal bundle) are satisfied.

7. Open Problems and Future Directions

Open cases remain regarding the equivalence of formal and absolute complete intersection properties in two dimensions and for nonintegral one-dimensional rings (Heitmann et al., 2011). Further, for lci curves and higher-codimension situations, structure theorems for the conormal bundle and its relationship to ideal-theoretic c.i. generation motivate continued paper, particularly in connection with non-excellent and singular phenomena.

In moduli theory, the local complete intersection property of monomial curves directly determines structure and smoothness of moduli spaces of Gorenstein curves with prescribed Weierstrass semigroup, making the lci criterion an essential combinatorial and geometric bridge (Contiero et al., 2022). The systematic numerical and homological criteria developed for both affine and projective settings connect the local intersection-theoretic behavior to global moduli properties and singularity‐theoretic constraints.