Profiles, linear spaces, and unirationality of complete intersections

Technical Summary: Profiles, Linear Spaces, and Unirationality of Complete Intersections

Problem Statement
In algebraic geometry, high-degree hypersurfaces and complete intersections in projective space are typically considered geometrically complex, often containing few or no rational subvarieties over the complex numbers. However, in positive characteristic $p > 0$, this correlation between degree and complexity breaks down. Notable examples, such as the Fermat hypersurface of degree $q+1$ (where $q=p^e$), exhibit unexpectedly simple behavior, including unirationality and the presence of many linear spaces, despite having high numerical degree. Traditional explanations for these phenomena often rely on ad hoc identifications with Hermitian forms. This work seeks to provide a systematic framework to explain why certain high-degree varieties in positive characteristic behave as if they were of low degree, specifically by analyzing the "special shape" of their defining polynomials.

Methodology and Framework
The paper introduces the concept of a profile, a discrete invariant associated with polynomials in positive characteristic that tracks their structural shape relative to a prime power $q = p^e$.

1. Profiles and $(q; a)$-tic Polynomials:
   A profile is an integer polynomial $a(t) = \sum a_j t^j$ with non-negative coefficients satisfying a technical injectivity condition (ensuring uniqueness in a specific tensor decomposition). A polynomial is called $(q; a)$-tic if it can be written as a sum of terms of the form $\prod f_{i,j}^{q^j}$, where each $f_{i,j}$ is homogeneous of degree $a_j$.
   The central heuristic of the paper is that $(q; a)$-tic hypersurfaces behave geometrically as if they were hypersurfaces of degree $a(1+\epsilon) \approx \sum a_j$ for a small $\epsilon > 0$, rather than their numerical degree $a(q) = \sum a_j q^j$.

2. Fano Schemes and Linear Spaces:
   The author studies the Fano schemes $F_r(X)$, which parameterize $r$-planes contained in a $(q; a)$-tic complete intersection $X$. By exploiting the tensor structure of the defining equations, the paper demonstrates that the Fano scheme is cut out by a section of a vector bundle whose rank depends on the profile $a$ rather than the numerical degree. This allows for precise dimension counts and smoothness results that are independent of $q$.

3. Inductive Unirationality Construction:
   To prove unirationality, the paper generalizes an old construction for cubic hypersurfaces (based on penultimate tangents) to the $(q; a)$-tic setting. The strategy involves:

   • Constructing a family of penultimate tangents (lines intersecting the variety with multiplicity $d-1$).
   • Defining a residual point map that sends a penultimate tangent to the remaining intersection point.
   • Using an inductive argument over a poset of multi-profiles. The induction proceeds by reducing the multi-profile via three operations: Frobenius descent (taking $q$-th roots), removing linear equations, and passing to the family of penultimate tangents (which reduces the profile).

Key Contributions and Results

• Theorem A (Geometry of Fano Schemes):
  The paper establishes that the Fano scheme $F_r(X)$ of $r$-planes in a general $(q; a)$-tic hypersurface $X \subset \mathbb{P}^n$ is determined entirely by the profile $a$.

  • If a specific numerical invariant $\delta(n, a, r)$ is negative, $F_r(X)$ is empty for general $X$.
  • If $\delta(n, a, r) \ge 0$, $F_r(X)$ is non-empty, irreducible, and of the expected dimension $\delta(n, a, r)$ for general $X$.
  • If the profile is reduced (constant term non-zero), $F_r(X)$ is smooth.
  • Crucially, these properties depend on $a$ and not on the prime power $q$, generalizing classical results for low-degree complete intersections.

• Theorem B (Unirationality):
  The main result states that for any multi-profile $a$, there exists an integer $n_0(a)$ such that for all $n \ge n_0(a)$, a general $(q; a)$-tic complete intersection in $\mathbb{P}^n$ is unirational.

  • This result holds regardless of the prime power $q$, provided the dimension $n$ is sufficiently large relative to the profile.
  • The proof utilizes the inductive reduction of profiles. The bound $n_0(a)$ is computed explicitly for small profiles.
  • For constant profiles $a(t) = d$ (corresponding to degree $d$ hypersurfaces), the bound $n_0(d)$ is significantly improved over previous classical bounds (e.g., $n_0(10) \approx 10^{59}$ compared to previous bounds of order $2^{10!}$).

• Conjecture C:
  The author conjectures that for smooth $(p; a)$-tic hypersurfaces with $a_0 \neq 0$, the Fano scheme $F_r(X)$ is irreducible of dimension $\delta(n, a, r)$ whenever $n$ is sufficiently large, uniformly across all characteristics $p > \max(a_i)$.

Significance and Claims
The paper claims to place historically isolated examples (like the Fermat hypersurface) into a broad, systematic context. By defining the "profile" of a polynomial, the work identifies a structural reason why certain high-degree varieties in positive characteristic are geometrically simple.

• Unification: It unifies the study of linear spaces and unirationality for a wide class of varieties that were previously treated as special cases.
• Quantitative Improvement: It provides significantly sharper bounds for the dimension required for unirationality in the classical case (constant profiles), improving upon results by Morin, Predonzan, and Ramanujam.
• Structural Insight: The work suggests that the "canonical sub-linear systems" arising from the profile structure are the primary obstruction to extending classical characteristic 0 results to positive characteristic, rather than the characteristic itself being a fundamental barrier.

The paper concludes by noting that while the current results are effective for large $n$, the techniques developed (particularly the global handling of families and the penultimate tangent construction) lay the groundwork for making these conditions effective for smaller dimensions and for exploring further properties like automorphisms and moduli.