[Paper Review] Interpolation for normal bundles of general curves

Atanas Atanasov, Eric Larson+1 more|arXiv (Cornell University)|Sep 5, 2015

Algebraic Geometry and Number Theory2 references3 citations

TL;DR

This paper establishes interpolation for the normal bundle $N_C$ of a general space curve of degree $d$ and genus $g$ in $\mathbb{P}^3$, using inductive techniques on degree and marked points. The key result is that $N_C$ satisfies interpolation unless $(d,g) = (6,1)$ or $(5,2)$, with the remaining case resolved via auxiliary induction and known results on lower-rank bundles.

ABSTRACT

Given n general points p_1, p_2,..., p_n in P^r, it is natural to ask when there exists a curve C \subset P^r, of degree d and genus g, passing through p_1, p_2,..., p_n. In this paper, we give a complete answer to this question for curves C with nonspecial hyperplane section. This result is a consequence of our main theorem, which states that the normal bundle N_C of a general nonspecial curve of degree d and genus g in P^r (with d >= g + r) has the property of interpolation (i.e. that for a general effective divisor D of any degree on C, either H^0(N_C(-D)) = 0 or H^1(N_C(-D)) = 0), with exactly three exceptions.

Motivation & Objective

• To determine when the normal bundle $N_C$ of a general space curve $C \subset \mathbb{P}^3$ satisfies interpolation.
• To resolve the interpolation problem for $N_C$ when $r=3$, extending prior results for rational curves.
• To characterize the exceptional cases where interpolation fails, particularly for $g > 0$.
• To establish a recursive framework using induction on degree and number of marked points to verify the interpolation condition.

Proposed method

• Uses induction on degree $d$ and number of marked points to verify that $N_C$ satisfies interpolation for $(d,g,3;n)$, assuming the result holds for smaller $d$ or fewer marked points.
• Applies Lemma \ref{two-secant} to reduce the problem to verifying goodness of a modified tuple $(d-1,g-1,3;n')$.
• Employs two key inductive lemmas: one for the case $\sum kn_{ij}^k = 1$, and another for $\sum kn_{ij}^k = 0$, each with distinct reduction rules.
• Relies on the regime condition $\sum (i-k)n_{ij}^k \leq 2d + 2g - 5$ to control the induction step.
• Uses auxiliary lemmas to handle exceptional configurations, such as points of type $(\ell,m;0)$ with $m \neq 0$, to avoid forbidden families.
• Combines results from lower-rank cases (e.g., $r=2$) and known theorems (e.g., for rational curves) to close the induction.

Experimental results

Research questions

• RQ1 For which triples $(d,g,3)$ does the normal bundle $N_C$ of a general space curve $C \subset \mathbb{P}^3$ satisfy interpolation?
• RQ2 What are the precise exceptions to interpolation when $g > 0$, and how can they be systematically excluded?
• RQ3 How can induction on degree and marked points be used to reduce the interpolation problem for $r=3$ to known or simpler cases?
• RQ4 What conditions on the tuple $(n_{ij}^k)$ ensure that the modified bundle $(d-1,g-1,3;n')$ remains good under the induction hypothesis?
• RQ5 How do the two infinite families of exceptions—defined by $\sum kn_{ij}^k = 0$ and $\sum in_{ij}^k = 2d+2g-14$, or $\sum kn_{ij}^k = 1$ and $\sum in_{ij}^k = 2d+2g-9$—affect the interpolation property?

Key findings

• The normal bundle $N_C$ satisfies interpolation for all general space curves in $\mathbb{P}^3$ unless $(d,g) = (6,1)$ or $(5,2)$, with the latter excluded by the assumptions of the main theorem.
• For $g > 0$, interpolation holds if $\sum (i-k)n_{ij}^k \leq 2d + 2g - 10$, which ensures the inductive reduction applies.
• The case $\sum kn_{ij}^k = 1$ is handled by reducing to $(d-1,g-1,3;n')$ with $\sum k(n')_{ij}^k = 0$, and verifying the new tuple avoids the forbidden families.
• When $\sum kn_{ij}^k = 0$ and $\sum in_{ij}^k = 2d+2g-14$, the existence of a point of type $(\ell,m;0)$ with $m \neq 0$ allows reduction to a tuple with strictly smaller $\sum in_{ij}^k$, enabling induction.
• The exceptional case $(d,g) = (6,1)$ is resolved by verifying that $(5,1,3;\mathbf{0})$ and $(5,1,2;\mathbf{0})$ are both good, relying on prior results for $r=2$ and $r=3$.
• The main theorem holds for $r=3$, completing the interpolation classification for normal bundles of general curves in $\mathbb{P}^3$.