Rational curves on cubic hypersurfaces in positive characteristic

1. Problem Statement

The paper investigates the geometry of the moduli spaces of rational curves on smooth cubic hypersurfaces defined over an algebraically closed field $k$ of characteristic $p \neq 2, 3$.

• Context: The study of rational curves on Fano varieties is central to classification theory (e.g., Minimal Model Program, rational connectedness). In characteristic 0, the irreducibility of these moduli spaces for cubic hypersurfaces of dimension $n \geq 4$ was established by Coskun and Starr [CS09].
• The Challenge: Extending these results to positive characteristic is non-trivial due to the existence of inseparable maps. In characteristic 0, the dominance of evaluation maps often implies the existence of free rational curves. In positive characteristic, this implication fails because a dominant map can be purely inseparable, meaning the generic fiber might not contain "free" curves (curves with ample normal bundles).
• Goal: To prove that for a smooth cubic hypersurface $X$ of dimension $n \geq 4$, the Kontsevich moduli space $\overline{M}_{0,0}(X, d)$ of stable maps of degree $d$ is irreducible and generically parametrizes free curves. Additionally, the paper addresses the case of cubic threefolds ($n=3$).

2. Methodology

Kitagawa employs a combination of dimension counting arguments, induction on the degree of the curves, and specific properties of cubic hypersurfaces in positive characteristic.

• Dimension Counting of Non-Free Loci:
  The core strategy involves bounding the dimension of the locus of "non-free" curves. The author utilizes a result from [Gla25] regarding the dimension of fibers of evaluation maps to perform an inductive dimension count. This allows the author to show that the locus of maps containing non-free components is a proper closed subset of the moduli space.
• Handling Inseparability:
  Instead of relying on the dominance of evaluation maps to deduce the existence of free curves (which fails in positive characteristic), the author explicitly calculates the upper bound of the dimension of the non-free locus. By showing this locus has strictly lower dimension than the expected dimension of the moduli space, the existence of free curves is guaranteed.
• Bend and Break Lemma:
  The proof utilizes the Bend and Break lemma to decompose high-degree stable maps into chains of lower-degree maps (specifically lines). This reduces the problem of irreducibility for degree $d$ to the irreducibility of fibers for lower degrees and the irreducibility of the space of lines.
• Gauss Map Analysis:
  For the base case of lines ($d=1$), the author analyzes the Gauss map $\gamma_X: X \to X^*$ to study the locus of non-free lines, proving it is a proper closed subset.

3. Key Contributions and Results

Main Theorems

• Theorem 1.1 (Dimension $n \geq 4$):
  Let $X$ be a smooth cubic hypersurface of dimension $n \geq 4$ over a field of characteristic $\neq 2, 3$. For every integer $d \geq 1$, the Kontsevich moduli space $\overline{M}_{0,0}(X, d)$ is irreducible and generically parametrizes free curves.

  • Significance: This extends the characteristic 0 result of Coskun and Starr to positive characteristic, overcoming the obstruction posed by inseparable maps.

• Theorem 1.2 (Dimension $n = 3$, Cubic Threefolds):
  Let $X$ be a smooth cubic threefold over a field of characteristic $\neq 2, 3$. For every integer $d \geq 1$, the moduli space $\overline{M}_{0,0}(X, d)$ generically parametrizes free curves.

  • Note: While the paper establishes the generic parametrization of free curves, it does not explicitly claim irreducibility for the entire space in the same manner as Theorem 1.1, likely due to the specific behavior of conics in dimension 3.

Specific Technical Results

• Irreducibility of Lines (Proposition 2.2): The space of lines $\overline{M}_{0,0}(X, 1)$ is smooth, irreducible, and generically parametrizes free lines.
• Dimension of Non-Free Locus (Lemma 3.3): For $n \geq 4$, the locus in $\overline{M}_{0,0}(X, d)$ parametrizing stable maps with at least one non-free component has dimension at most $d(n-1)$. Since the expected dimension of the moduli space is $d(n-1) + (n-3)$ (or similar depending on the specific count), the non-free locus is a proper subset.
• Cubic Threefolds and Conics (Lemma 4.1): For $n=3$, the space of conics $\overline{M}_{0,0}(X, 2)$ consists of two irreducible components:
  1. $R_2$: Birational maps from irreducible curves to smooth conics.
  2. $N_2$: Degree 2 covers from $\mathbb{P}^1$ to lines.
     Both components generically parametrize free curves.

Counter-Example and Limitations

• Characteristic 2: The results do not hold for the Fermat cubic hypersurface in characteristic 2. In this specific case, the hypersurface is not globally $F$-regular, and all lines on it are non-free. This highlights the necessity of the characteristic $\neq 2$ assumption.

4. Significance and Implications

• Geometric Manin's Conjecture: The results provide new examples supporting the Geometric Manin's Conjecture (proposed by Lehmann and Tanimoto) in positive characteristic. This conjecture predicts the structure of moduli spaces of rational curves (number of components, dimensions, etc.). The paper confirms that for cubic hypersurfaces, the "main" components are separable and parametrized by free curves, aligning with the conjecture's predictions.
• Overcoming Inseparability: The paper demonstrates a robust method for handling inseparable maps in positive characteristic by using dimension counting rather than relying solely on the properties of evaluation maps. This technique is likely applicable to other Fano varieties in positive characteristic.
• Generalization of [CS09]: It successfully generalizes the landmark result of Coskun and Starr from characteristic 0 to positive characteristic (excluding 2 and 3), filling a significant gap in the classification of rational curves on Fano varieties.

Summary

Kitagawa's paper resolves the irreducibility and freeness of rational curves on high-dimensional cubic hypersurfaces in positive characteristic. By carefully analyzing the dimension of non-free loci and utilizing the geometry of the Gauss map, the author proves that despite the presence of inseparable maps, the moduli spaces behave "generically" as they do in characteristic 0: they are irreducible and dominated by free curves. This work strengthens the theoretical foundation for Geometric Manin's Conjecture in positive characteristic settings.