On a counterexample to a conjecture of J. Harris for octic surfaces

Technical Summary: On a Counterexample to a Conjecture of J. Harris for Octic Surfaces

Problem Statement
The paper addresses a conjecture by J. Harris (1980s) regarding the Noether-Lefschetz locus $NL_d$ in the parameter space $T$ of smooth surfaces of degree $d \geq 4$ in $\mathbb{P}^3$. Harris conjectured that the number of "special" components of $NL_d$ (those with codimension strictly less than the general codimension $h^{2,0} = \binom{d-1}{3}$) must be finite. While counterexamples were found by C. Voisin for large $d$, the conjecture remains open for lower degrees. Specifically, for $d=6$ and $d=7$, the number of reduced special components is known to be finite. The focus of this work is the case $d=8$ (octic surfaces), where the general codimension is 35. Previous work by the author ([Mov21b]) suggested the existence of an infinite number of components of codimension 31, which would contradict Harris' conjecture. This paper aims to gather stronger evidence for this claim.

Methodology
The author investigates the local Noether-Lefschetz locus $V_{\delta_0}$ associated with a specific cohomology class $\delta_0 = [C_1] + r[C_2]$ on the Fermat octic surface $X_0$. Here, $C_1$ is a line (complete intersection of type $(1,1)$) and $C_2$ is a complete intersection of type $(3,3)$, chosen such that they do not intersect. The parameter $r$ is a non-zero rational number.

The methodology involves:

1. Geometric Setup: Defining the deformation space $T$ of smooth octic surfaces and specific subspaces $\check{T}$ and $\check{S}$ with reduced dimensions to facilitate computation.
2. Period Integrals and Hodge Theory: Utilizing the Gauss-Manin connection and residue maps to express the condition that a cohomology class remains of type $(1,1)$ as the vanishing of period integrals $\int_{\delta_t} \omega_\beta$.
3. Smoothness Analysis: Investigating the "smoothness" of the Noether-Lefschetz locus. A locus is defined as $N$-smooth if its infinitesimal approximation up to order $N$ corresponds to a smooth variety. The paper proves smoothness for various orders ($N=2, 5, 6, 9$) on different subspaces.
4. Tangent Space Computations: Calculating the codimensions of tangent spaces $T_0 V$ for the loci associated with $C_1$, $C_2$, and their linear combination.
5. Computer Algebra: Extensive computations are performed using the Singular computer algebra system (specifically the foliation.lib library) to calculate ranks of matrices derived from period integrals and to verify smoothness conditions up to high orders.
6. LLM-Assisted Discovery (and Verification): The author notes that a Large Language Model (LLM) was initially used to identify a specific 11-dimensional subspace $\check{S}$ where higher-order smoothness could be verified, though the final proofs and code were generated independently by the author.

Key Contributions and Results

• Conjecture 1: The paper proposes that for all but a finite number of $r \in \mathbb{Q}$, the local Noether-Lefschetz loci $V_{[C_1]+r[C_2]}$ are smooth, distinct analytic spaces of codimension 31. These loci intersect each other in a common subvariety of codimension 32.
• Codimension Analysis:
  • The general components of $NL_8$ have codimension 35.
  • The components associated with $C_1$ (lines) have codimension 5.
  • The components associated with $C_2$ (type $(3,3)$ curves) have codimension 27.
  • The components associated with the sum $[C_1] + r[C_2]$ are shown to have codimension 31 in the full parameter space $T$.
• Smoothness Proofs:
  • Proposition 3: Proves that $V_{[C_1]+r[C_2]}$ is 2-smooth in the full parameter space $T$.
  • Proposition 4: Proves that the restriction to a specific subspace $\check{T}$ (dimension 149) yields a general pencil, with tangent space codimensions of 9 for the sum and 10 for the intersection.
  • Proposition 5: Demonstrates that on a specific 11-dimensional subspace $\check{S}$, the locus is 6-smooth (for a family with 10 monomials) and 9-smooth (for a family with a single monomial).
• Implication of Infinite Components: By combining Proposition 4 and Proposition 2 (which relates the geometry of the tangent spaces to the number of irreducible components), the author concludes that the loci $V_{[C_1]+r[C_2]}$ for varying $r$ possess an infinite number of irreducible components of codimension $\geq 31$.
• Counterexample Potential: Since the maximum codimension for components in this specific case is 35 (the general case), and the author provides evidence for components of codimension 31, the existence of these components would constitute a counterexample to Harris' conjecture for $d=8$.

Significance
The paper provides "strong evidences" rather than a definitive proof that Harris' conjecture fails for octic surfaces ($d=8$). The significance lies in:

1. Challenging the Finiteness Conjecture: It suggests that the number of special components for $d=8$ is infinite, contrary to the expectation that Harris' conjecture holds for low degrees.
2. Geometric Structure: It elucidates the structure of the Noether-Lefschetz locus for specific algebraic cycles, showing that they form a pencil of distinct, smooth, codimension-31 subvarieties intersecting in a codimension-32 variety.
3. Methodological Integration: The work demonstrates the utility of combining classical Hodge theory, algebraic geometry, and computational algebra (Singular) to tackle high-dimensional moduli problems.

The author remains cautious, noting that it is still possible these loci lie inside a larger irreducible component of codimension less than 31, though this is considered a "hard question." The paper does not claim to have definitively disproven Harris' conjecture but presents a robust framework and computational evidence suggesting it is false for $d=8$.