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Initial ideals of weighted forms and the genus of locally Cohen-Macaulay curves

Alessio Sammartano, Enrico Schlesinger

TL;DR

The paper resolves the sharpness of the upper bound for the maximum genus of locally Cohen-Macaulay curves in P^3 by translating the geometric question into the weighted-algebraic problem of initial ideals in a nonstandard graded ring with weights wt(x)=1,wt(y)=2,wt(z)=3. It proves that for a general wt-homogeneous form f of weight 3m, the multiplication map f·:R[−3m]→M is an isomorphism, which is equivalent to the initial ideal of (x,y)^{3m−2}+(f) being (x,y,z)^{3m−2}. The core method constructs a canonical bijection φ between monomial bases, decomposing into rectangular and triangular regions, and proving divisibility and, in favorable weights, uniqueness; for other weights, a refined Non Cancellation Lemma still yields the required isomorphism. The results confirm Beorchia–Lella–Sammartano’s conjecture in the d=s and d≥2s−1 cases, opening a path to the maximum genus predictions and linking algebraic combinatorics of weighted initial ideals to geometric genus bounds. These findings bridge weighted Gröbner-type questions with classical curve-genus problems, providing a rigorous combinatorial route to sharp genus bounds and offering explicit obstructions and remedies via special blocks and non-cancellation techniques.

Abstract

Let C be a locally Cohen-Macaulay curve in complex projective 3-space. The maximum genus problem predicts the largest possible arithmetic genus g(d,s) that C can achieve assuming that it has degree d and does not lie on surfaces of degree less than s. In this paper, we prove that this prediction is correct when d=s or d is at least 2s-1. We obtain this result by proving another conjecture, by Beorchia, Lella, and the second author, about initial ideals associated to certain homogeneous forms in a non-standard graded polynomial ring.

Initial ideals of weighted forms and the genus of locally Cohen-Macaulay curves

TL;DR

The paper resolves the sharpness of the upper bound for the maximum genus of locally Cohen-Macaulay curves in P^3 by translating the geometric question into the weighted-algebraic problem of initial ideals in a nonstandard graded ring with weights wt(x)=1,wt(y)=2,wt(z)=3. It proves that for a general wt-homogeneous form f of weight 3m, the multiplication map f·:R[−3m]→M is an isomorphism, which is equivalent to the initial ideal of (x,y)^{3m−2}+(f) being (x,y,z)^{3m−2}. The core method constructs a canonical bijection φ between monomial bases, decomposing into rectangular and triangular regions, and proving divisibility and, in favorable weights, uniqueness; for other weights, a refined Non Cancellation Lemma still yields the required isomorphism. The results confirm Beorchia–Lella–Sammartano’s conjecture in the d=s and d≥2s−1 cases, opening a path to the maximum genus predictions and linking algebraic combinatorics of weighted initial ideals to geometric genus bounds. These findings bridge weighted Gröbner-type questions with classical curve-genus problems, providing a rigorous combinatorial route to sharp genus bounds and offering explicit obstructions and remedies via special blocks and non-cancellation techniques.

Abstract

Let C be a locally Cohen-Macaulay curve in complex projective 3-space. The maximum genus problem predicts the largest possible arithmetic genus g(d,s) that C can achieve assuming that it has degree d and does not lie on surfaces of degree less than s. In this paper, we prove that this prediction is correct when d=s or d is at least 2s-1. We obtain this result by proving another conjecture, by Beorchia, Lella, and the second author, about initial ideals associated to certain homogeneous forms in a non-standard graded polynomial ring.
Paper Structure (18 sections, 29 theorems, 141 equations, 6 figures, 4 tables)

This paper contains 18 sections, 29 theorems, 141 equations, 6 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction of size for the matrix $\mathbf{A}_w$
  4. Construction of the bijection $\varphi$ satisfying divisibility - part one
  5. The bijection $\varphi^\hrectangle$ between the rectangular regions
  6. An order relation on the rectangular regions
  7. Uniqueness in the rectangular regions
  8. Construction of the bijection $\varphi$ satisfying divisibility - part two
  9. Technical results on the triangular regions
  10. Uniqueness in the triangular regions
  11. The bijection $\varphi: \mathscr{R}'_w \rightarrow \mathscr{M}'_w$ and its uniqueness for $w \equiv 0$ modulo $3$
  12. Examples of non-uniqueness
  13. Non Cancellation Lemma
  14. The cases $w \equiv 1$ or $2$ modulo $3$
  15. Special blocks
  16. ...and 3 more sections

Key Result

Theorem 1.1

Conjectures Conj1 and Conj2 are true. As a consequence, if either $d=s \geq 1$ or $d \geq 2s+1 \geq 3$, the maximum genus of a locally Cohen-Macaulay curve in $\mathbb{P}^3_{\mathbb{C}}$ of degree $d$ that does not lie on a surface of degree $s-1$ is equal to $g(d,s)$.

Figures (6)

  • Figure 1: The map $\varphi$ for $m=7$ and $w=18$
  • Figure 2: The map $\varphi$ for $m=7$ and $w=19$
  • Figure 3: The map $\varphi$ for $m=7$ and $w=20$
  • Figure 4: The map $\varphi$ for $m=7$ and $w=21$
  • Figure 5: The map $\varphi$ for $m=7$ and $w=22$
  • ...and 1 more figures

Theorems & Definitions (79)

  • Conjecture 1: BLS
  • Conjecture 2
  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof
  • Proposition 2.3
  • Proposition 3.1
  • ...and 69 more