On complete intersection projective closures of monomial curves

Technical Summary: Complete Intersection Projective Closures of Monomial Curves

Problem Statement
This paper investigates the conditions under which the projective closure of an affine monomial curve is a complete intersection. While the complete intersection property for affine monomial curves is well-established, the behavior of their projective closures remains less understood. The central difficulty arises because the homogenization of the defining toric ideal of an affine curve can increase the minimal number of generators and alter the structure of the defining equations. Consequently, an affine monomial curve being a complete intersection is a necessary but not sufficient condition for its projective closure to share this property. The paper specifically targets monomial curves in four-dimensional affine space ($n=4$), aiming to provide a complete and explicit characterization of when their projective closures are complete intersections and when they are arithmetically Cohen–Macaulay.

Methodology
The analysis relies on the structural classification of minimal binomial generating sets for the affine toric ideal $I(\mathbf{a})$ of a monomial curve $C(\mathbf{a})$ in $\mathbb{A}^4$. The author utilizes the degree reverse lexicographic order and Gröbner basis techniques to analyze the homogenized ideal $I_h(\mathbf{a})$.

Key methodological steps include:

1. Structural Classification: Leveraging Theorem 2.3, the minimal generating set of $I(\mathbf{a})$ is categorized into three distinct cases based on the relations between the integers $c_i$ (the smallest positive integers such that $c_i a_i \in \sum_{j \neq i} \mathbb{N}a_j$).
2. Gröbner Basis Analysis: For each structural case, the paper constructs specific sets of binomials and verifies whether they form a Gröbner basis for $I(\mathbf{a})$ and, subsequently, for $I_h(\mathbf{a})$.
3. Initial Monomial Analysis: The paper applies Buchberger's criterion and properties of initial monomials. Specifically, it utilizes the fact that if the initial monomials of a generating set are pairwise relatively prime, the set forms a Gröbner basis.
4. Arithmetically Cohen–Macaulay (ACM) Characterization: The paper employs criteria from existing literature (specifically [6, Theorem 2.2]) which link the ACM property to the non-divisibility of initial monomials by specific variables (typically $x_4$ or $x_1$ depending on the ordering).

Key Contributions and Results

The paper provides necessary and sufficient conditions for the projective closure $C(\mathbf{a}) \subset \mathbb{P}^4$ to be a complete intersection and/or arithmetically Cohen–Macaulay, divided by the three structural cases of the affine ideal:

• Case 1: The affine ideal is generated by $\{x_2^{c_2} - x_1^{c_1}, x_3^{c_3} - x_4^{c_4}, x_1^{d_1}x_2^{d_2} - x_3^{d_3}x_4^{d_4}\}$.

  • Complete Intersection: Theorem 2.4 and 2.6 establish that $I_h(\mathbf{a})$ is a complete intersection if $d_2=c_2, d_3=c_3$ or if $d_2=0, d_3=c_3$.
  • ACM Property: Theorems 2.7, 2.9, 2.10, and 2.12 provide explicit inequalities involving the exponents $c_i$ and $d_i$ that are necessary and sufficient for the curve to be arithmetically Cohen–Macaulay.
  • Corollary: Corollaries 2.8 and 2.13 demonstrate that in specific subcases, the ACM property is equivalent to the complete intersection property.

• Case 2: The affine ideal is generated by $\{x_2^{c_2} - x_1^{c_1}, x_3^{c_3} - x_1^{c_1}, x_4^{c_4} - x_1^{d_1}x_2^{d_2}x_3^{d_3}\}$.

  • Complete Intersection: Theorems 3.1, 3.8, and 3.12 provide criteria based on the relative sizes of exponents (e.g., $d_2=c_2$ and $d_3=c_3$) and factorization properties of the integers $c_i a_i$.
  • ACM Property: Theorems 3.3 and 3.6 characterize the ACM property in subcases where $d_2 < c_2$ or $d_3 < c_3$.
  • Infinite Families: Proposition 3.11 constructs infinite families of monomial curves where the projective closure is arithmetically Cohen–Macaulay, yet the homogeneous toric ideal is not a complete intersection and possesses an arbitrarily large number of minimal generators ($2n+2$). This explicitly demonstrates that the ACM property does not imply the complete intersection property.

• Case 3: The affine ideal is generated by $\{x_2^{c_2} - x_1^{c_1}, x_3^{c_3} - x_1^{b_1}x_2^{b_2}, x_4^{c_4} - x_1^{d_1}x_2^{d_2}x_3^{d_3}\}$.

  • Complete Intersection: Theorems 4.1, 4.3, 4.5, 4.7, 4.11, and 4.13 derive explicit criteria. These conditions often involve the existence of specific factorizations of the integers $c_3 a_3$ and $c_4 a_4$ into linear combinations of the sequence $\mathbf{a}$, subject to constraints on the sum of coefficients.
  • Subcases: The results distinguish between scenarios where $a_1$ is the maximum element, $a_3$ is the maximum, or $a_4$ is the maximum, and further subdivide based on the relationships between $c_3, b_1, b_2$ and $c_4, d_1, d_2, d_3$.

Significance and Claims
The paper claims to provide the first complete and explicit characterization of the complete intersection property for projective closures of complete intersection affine monomial curves in four-dimensional affine space. Previous works (e.g., [1], [2]) addressed only specific families or simplicial cases, leaving a gap in the general classification.

The author emphasizes that the approach combines detailed generator analysis with Gröbner basis techniques to resolve the structural changes induced by homogenization. A significant finding is the construction of families where the projective closure is arithmetically Cohen–Macaulay but fails to be a complete intersection, highlighting the independence of these two algebraic properties in the projective setting.

The paper acknowledges that while a full characterization for the arithmetically Cohen–Macaulay property is achieved in Case 1 and partially in Case 2, the full characterization for Cases 2 and 3 remains an open problem. The results are presented as explicit criteria expressed in terms of the minimal binomial generators of the affine toric ideal, offering a concrete tool for determining the geometric and algebraic properties of these curves.