Interpolation for rational curves with secants

Technical Summary: Interpolation for Rational Curves with Secants

Problem Statement
This paper addresses the problem of interpolation for rational curves in projective space $\mathbb{P}^r$ subject to a secancy condition. Specifically, given integers $d, k, r, s$ and a linear space $P_s \subset \mathbb{P}^r$, the authors investigate the existence and enumeration of rational curves $f: \mathbb{P}^1 \to \mathbb{P}^r$ of degree $d$ that meet $P_s$ with multiplicity $k$ and pass through $n$ general points $x_1, \dots, x_n$. The study considers two scenarios:

1. Unprescribed points: Determining the maximum number of general points a general such curve can pass through (Question 1.1.1(a)).
2. Prescribed points: Determining the maximum number of general points $p_i \in \mathbb{P}^1$ such that $f(p_i) = x_i$ for general $x_i$ (Question 1.1.1(b)).

Geometrically, these questions are reformulated in terms of the blow-up $X_{r,s} = \text{Bl}_{P_s}(\mathbb{P}^r)$. A curve meeting $P_s$ with multiplicity $k$ corresponds to a map $f: \mathbb{P}^1 \to X_{r,s}$ of degree $(d, k)$. The questions reduce to determining whether the evaluation maps from the moduli space of such maps to the target space are dominant.

Methodology
The authors determine the answers by analyzing the splitting types of two specific vector bundles associated with a general map $f$:

• The normal bundle $N_{C/X_{r,s}}$, which governs the deformation of the curve within the ambient space (related to unprescribed points).
• The restricted tangent bundle $f^* T_{X_{r,s}}$, which governs the deformation of the map with fixed domain points (related to prescribed points).

The central technical tool is the property of balancedness. A vector bundle on $\mathbb{P}^1$ is balanced if it splits as a direct sum of line bundles with degrees differing by at most 1. In characteristic 0, a bundle is balanced if and only if it satisfies the interpolation property (i.e., sections vanish at general points as expected). The authors extend these results to arbitrary characteristic, including characteristic 2, where additional obstructions arise.

The primary methodological approach is degeneration. The authors construct singular nodal curves $C = C' \cup L$ (where $L$ is a line of specific degree) and analyze the normal and tangent bundles of these singular curves. Using the theory of elementary modifications and pointing bundles, they relate the bundles of the singular curve to those of the smooth components. By proving that interpolation is an open condition in flat families, they deduce the properties of the general smooth curve from the singular ones.

Key technical steps include:

• Inductive Steps: Establishing that if a bundle for degree $(d, k)$ is balanced (or "almost balanced"), then the bundle for $(d + r - s - 1, k + 1)$ is also balanced. This involves adding specific configurations of lines to the curve.
• Obstruction Analysis: Identifying specific geometric and arithmetic conditions under which balancedness fails. These include:
  • Degeneracy: When the image of the curve lies in a proper linear subspace.
  • Projection Sequences: When the slope of the relative tangent bundle is too large compared to the normal bundle of the projection, causing the extension sequence to split in an unbalanced way.
  • Characteristic 2 Obstructions: Specific failures in characteristic 2 related to the Frobenius map and the parity of degrees, which prevent balancedness even when slope conditions are met.

Key Contributions and Results

1. Complete Determination of Normal Bundles:
   The paper provides a complete classification of the splitting type of the normal bundle $N_{C/X_{r,s}}$ for general rational curves in $X_{r,s}$ in arbitrary characteristic.

   • Characteristic $\neq 2$: The bundle is balanced if and only if the curve is non-degenerate and the degree $d$ satisfies a specific lower bound $d \ge L_{r,s}(k)$. If $d$ is smaller, the bundle splits as a direct sum of the relative tangent bundle and the normal bundle of the projection, which is unbalanced.
   • Characteristic 2: The authors identify that balancedness fails not only due to the projection sequence splitting but also due to obstructions inherited from the ambient projective space $\mathbb{P}^r$ (specifically when $(r-1) \nmid (d-1)$) and specific arithmetic conditions on $d$ and $k$. Theorem 1.4.3 states that these are the only obstructions.

2. Complete Determination of Restricted Tangent Bundles:
   The paper determines the splitting type of the restricted tangent bundle $f^* T_{X_{r,s}}$ in all characteristics.

   • Unlike the normal bundle, the restricted tangent bundle is balanced if and only if a specific inequality involving $d, k, r, s$ holds ($d \ge L^t_{r,s}(k)$). This condition is uniform across all characteristics.
   • This result implies that the maximum number of prescribed points a general curve can pass through is determined uniformly by the slope of the restricted tangent bundle, regardless of the field characteristic.

3. Enumerative Results (Tevelev Degrees):
   In the appendix, the authors enumerate the number of such interpolating curves when the number of points is maximal. They compute the "Tevelev degrees" for $X_{r,s}$, providing a formula for the number of maps satisfying $n$ general incidence conditions. This recovers previous results for $s=0$ and generalizes them to higher-dimensional linear spaces.

Significance and Claims
The paper claims to resolve the interpolation problem for rational curves with secants in arbitrary characteristic. Its significance lies in:

• Uniformity: Establishing that the answer to the prescribed point problem (Question 1.1.1(b)) is uniform in characteristic, whereas the unprescribed point problem (Question 1.1.1(a)) exhibits distinct behavior in characteristic 2.
• Completeness: Providing a full list of obstructions to balancedness, showing that in characteristic 2, the failure of balancedness is entirely explained by the splitting of the projection sequence or the propagation of unbalancedness from the ambient projective space.
• Generalization: Extending known results for rational curves in $\mathbb{P}^r$ (where $k=0$) to the more complex setting of curves with secancy conditions ($k>0$) on blow-ups of projective spaces.

The authors explicitly note that while they focus on rational curves, the methods and questions are relevant to general curves of positive genus, though those cases are not treated in this work. The results are derived rigorously using algebraic geometry techniques, specifically moduli spaces, vector bundle theory, and degeneration arguments, without reliance on experimental data or unproven conjectures.