[Paper Review] Lines on Fermat surfaces

Matthias Schuett, Tetsuji Shioda+1 more|ArXiv.org|Dec 12, 2008

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper proves that the Néron-Severi group of the complex Fermat surface of degree $m$ is integrally generated by lines if and only if $m \leq 4$ or $m$ is coprime to 6, for all $m \leq 100$. The proof uses supersingular reduction modulo a prime $p$, comparing discriminants of Néron-Severi lattices in characteristic $p$ with those of the integral lattice generated by lines over $\mathbb{C}$, leveraging the Artin-Tate conjecture and computational verification via MAGMA.

ABSTRACT

We prove that the Neron-Severi groups of several complex Fermat surfaces are generated by lines. Specifically, we obtain these new results for all degrees up to 100 that are relatively prime to 6. The proof uses reduction modulo a supersingular prime. The techniques are developed in detail. They can be applied to other surfaces and varieties as well.

Motivation & Objective

• To determine for which degrees $m \leq 100$ the Néron-Severi group of the complex Fermat surface $S$ of degree $m$ is integrally generated by the $3m^2$ obvious lines.
• To resolve a long-standing question about integral generation of $\mathop{\rm NS}(S)$ by lines, extending prior results on rational generation.
• To develop and apply a refined supersingular reduction technique that avoids reliance on multiple reductions, enabling systematic computation for higher degrees.
• To provide a computational framework using MAGMA to verify the injectivity of lattice maps over finite fields, ensuring discriminant comparisons are valid.

Proposed method

• Use supersingular reduction modulo a prime $p$ to relate the Néron-Severi lattice $\mathop{\rm NS}(S)$ over $\mathbb{C}$ to $\mathop{\rm NS}(S_p)$ in positive characteristic.
• Construct a sublattice $L \subset \mathop{\rm NS}(S_p)$ generated by reductions of the $3m^2$ lines and additional divisors arising in characteristic $p$, particularly from the action of $\mu_m^4 / \mu_m$.
• Compare the discriminant of the lattice $M$ generated by lines over $\mathbb{Z}$ with the discriminant of $L$ in $\mathop{\rm NS}(S_p)$, using the relation $\mathop{\rm disc}(L) = [\Lambda:L]^2 \cdot \mathop{\rm disc}(\Lambda)$ for sublattices.
• Apply Criterion 5.3: if the discriminants of $M$ and $L$ do not share a common square factor, then $M$ must be saturated in $\mathop{\rm NS}(S)$, implying integral generation.
• Use the Artin-Tate conjecture to control the discriminant of $\mathop{\rm NS}(S_p)$, especially in the supersingular case where the discriminant is well-understood.
• Implement the method computationally using MAGMA, computing intersection forms modulo $\ell$ and testing injectivity of the map $M_{\mathbb{F}_\ell} \to L_{\mathbb{F}_\ell}^*$ via kernel computations with random orbits of divisors.

Experimental results

Research questions

• RQ1 For which degrees $m \leq 100$ is the Néron-Severi group of the complex Fermat surface $S$ of degree $m$ integrally generated by the $3m^2$ obvious lines?
• RQ2 Can the supersingular reduction technique be extended to avoid reliance on multiple reductions, as in earlier Picard number computations?
• RQ3 To what extent do the discriminants of the Néron-Severi lattices in characteristic $p$ and over $\mathbb{C}$ constrain the integral structure of $\mathop{\rm NS}(S)$?
• RQ4 How can computational algebra systems be used effectively to verify injectivity of lattice maps modulo primes without full matrix computation?
• RQ5 What role do the orbits of special lines under $\mu_m^4 / \mu_m$ play in completing the Néron-Severi lattice in positive characteristic?

Key findings

• The Néron-Severi group of the complex Fermat surface of degree $m$ is integrally generated by lines if and only if $m \leq 4$ or $\gcd(m,6) = 1$, for all $m \leq 100$.
• For $m \leq 100$ with $\gcd(m,6) \neq 1$ and $m > 4$, the Néron-Severi group is not even rationally generated by lines, so integral generation fails.
• The discriminant of the lattice $M$ generated by the $3m^2$ lines depends only on $m$, and is divisible only by primes dividing $m$, as shown in Corollary 4.4.
• The method successfully verified integral generation for all $m \leq 100$ with $\gcd(m,6) = 1$ by confirming that the map $M_{\mathbb{F}_\ell} \to L_{\mathbb{F}_\ell}^*$ is injective for some $\ell$ dividing $m$, via computational checks on kernel intersections.
• The implementation used random selection of lines and their orbits under $\mu_m^4 / \mu_m$, with incremental kernel computation to avoid memory overflow, and succeeded in all cases with at most two additional divisors.
• The technique is generalizable to other surfaces and varieties, as demonstrated by the extension of the supersingular reduction method in section 7.4, which reduces computational cost while maintaining correctness.