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arxiv: 2607.01432 · v1 · pith:RMHDIPE2new · submitted 2026-07-01 · 🧮 math.AG

Interpolation for rational curves with secants

Alessio Cela , Carl Lian This is my paper

Pith reviewed 2026-07-03 18:26 UTC · model grok-4.3

classification 🧮 math.AG
keywords rational curvesinterpolationsecant conditionsblowupsnormal bundlesprojective spacealgebraic geometry
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The pith

Rational curves of degree d in P^r pass through a maximum number of general points when also secant to a linear space, and this number is computed in any characteristic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper determines the largest number of general points that a rational curve of degree d in projective r-space can pass through when it must also meet a linear space along a secant. It treats both the case where the points of intersection with the curve remain free and the case where those points are fixed in advance. These numbers are obtained by computing the normal bundle and the restricted tangent bundle of a general rational curve inside the blowup of projective space along the linear space. The results apply over fields of arbitrary characteristic. A sympathetic reader would care because the computation supplies the complete interpolation count for rational curves under this linear secancy constraint.

Core claim

In arbitrary characteristic, the maximum number of general points through which a rational curve of degree d in P^r passes is determined subject to an additional secancy condition along a linear space. The cases of unprescribed and prescribed points on the curve correspond to the normal and restricted tangent bundles of a general rational curve in the blowup Bl_{P^s} P^r, respectively. The appendix enumerates the actual interpolating curves when the points are prescribed.

What carries the argument

The normal bundle and restricted tangent bundle of a general rational curve in Bl_{P^s} P^r, from which the interpolation numbers under the secancy condition are read off.

If this is right

  • The maximum holds uniformly in every characteristic.
  • Unprescribed-point counts are given by the normal bundle on the blowup.
  • Prescribed-point counts are given by the restricted tangent bundle on the blowup.
  • The appendix supplies the finite number of such curves when the intersection points are fixed in advance.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same blowup-and-bundle technique could be applied to count rational curves meeting several linear spaces simultaneously.
  • The resulting dimension formulas might be tested by direct enumeration in small values of r and d.
  • The method suggests a route to interpolation statements for rational curves subject to other linear incidence conditions.

Load-bearing premise

The generality of the points and the secancy condition permit the counts to be read off from the normal and restricted tangent bundles of a general rational curve in the blowup.

What would settle it

A rational curve of degree d in P^r that remains secant to the linear space yet passes through strictly more general points than the number obtained from the bundle computation would falsify the stated maximum.

read the original abstract

In arbitrary characteristic, we determine the maximum number of general points through which a rational curve of degree $d$ in $\mathbb{P}^r$ passes, subject to an additional secancy condition along a linear space. We consider the cases both where the points on the curve are unprescribed and prescribed, which amount to the determination of the normal and restricted tangent bundles of a general rational curve in $\mathsf{Bl}_{\mathbb{P}^s}\mathbb{P}^r$, respectively. In the appendix, we enumerate the interpolating curves in the case of prescribed points on the curve.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to determine, in arbitrary characteristic, the maximum number of general points through which a rational curve of degree d in P^r passes subject to a secancy condition along a linear space P^s. The unprescribed and prescribed cases are reduced to computing the normal bundle and restricted tangent bundle, respectively, of a general rational curve in the blowup Bl_{P^s} P^r; the appendix enumerates the interpolating curves in the prescribed case.

Significance. If the central claims hold, the work supplies explicit interpolation counts for rational curves with secant conditions that are valid in all characteristics. This extends classical results on rational curve interpolation and addresses potential characteristic-dependent behavior in moduli and cohomology, with the blowup construction providing a uniform framework and the appendix supplying concrete enumerative data.

major comments (2)
  1. [Abstract and introduction] Abstract and introduction: the determination of the maximum counts is asserted to follow directly from h^0 of the normal bundle (unprescribed) and restricted tangent bundle (prescribed) on a general rational curve in Bl_{P^s} P^r. The manuscript must supply an explicit argument that this general member achieves the maximum in every characteristic, including when the map P^1 → Bl_{P^s} P^r is inseparable, and that no special curve in char p > 0 passes through strictly more points.
  2. [Main body (bundle computations)] The reduction to bundle cohomology assumes that the dimension of the family equals h^0 minus the expected obstructions with no positive-dimensional stabilizers. A verification that these vanishings and dimension counts remain valid after base change to positive characteristic is required, as the generality assumption for the curve in the blowup may fail to capture the maximum when characteristic-dependent conditions arise on the exceptional divisor.
minor comments (2)
  1. Clarify the precise range of d, r, s for which the stated generality of the rational curve holds.
  2. [Appendix] Ensure the appendix enumeration is cross-referenced with the bundle computations to confirm agreement of the counts.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying these points on the characteristic-independent aspects of the argument. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and introduction] Abstract and introduction: the determination of the maximum counts is asserted to follow directly from h^0 of the normal bundle (unprescribed) and restricted tangent bundle (prescribed) on a general rational curve in Bl_{P^s} P^r. The manuscript must supply an explicit argument that this general member achieves the maximum in every characteristic, including when the map P^1 → Bl_{P^s} P^r is inseparable, and that no special curve in char p > 0 passes through strictly more points.

    Authors: We agree that an explicit argument for maximality in positive characteristic, including the inseparable case, should be supplied rather than left implicit. The bundle computations themselves are performed via exact sequences whose splitting types are determined generically and are independent of characteristic. In the revised manuscript we will add a short subsection (likely after the statement of the main theorems) showing that the dimension of the space of sections is upper semicontinuous under specialization and that any curve achieving a strictly larger count would force a jump in cohomology that contradicts the generality of the chosen rational curve in the blowup. This addresses both the inseparable maps and the comparison with special curves. revision: yes

  2. Referee: [Main body (bundle computations)] The reduction to bundle cohomology assumes that the dimension of the family equals h^0 minus the expected obstructions with no positive-dimensional stabilizers. A verification that these vanishings and dimension counts remain valid after base change to positive characteristic is required, as the generality assumption for the curve in the blowup may fail to capture the maximum when characteristic-dependent conditions arise on the exceptional divisor.

    Authors: The exact sequences used to compute the normal and restricted tangent bundles hold over any base field, and the resulting negative summands that produce the vanishings of H^1 are likewise characteristic-independent. Nevertheless, we acknowledge that an explicit check that the chosen general curve avoids any characteristic-dependent loci on the exceptional divisor is not written out. In revision we will insert a paragraph verifying that the open set of curves whose normal bundle splits in the expected way remains nonempty after base change, using the fact that the exceptional divisor is a projective space and the incidence conditions are linear. This will confirm that the dimension count equals h^0 minus obstructions with no positive-dimensional stabilizers in all characteristics. revision: yes

Circularity Check

0 steps flagged

No circularity; standard reduction to bundle cohomology with no self-referential definitions or load-bearing self-citations

full rationale

The paper equates the maximum interpolation count (with or without prescribed points) to the dimensions of global sections of the normal bundle and restricted tangent bundle of a general rational curve in the blowup. This is a direct, non-circular reduction standard in algebraic geometry: the count is computed from the cohomology rather than being defined in terms of itself. No equations, ansatzes, or predictions are shown to reduce by construction to fitted inputs or prior self-citations. The abstract states the equivalence explicitly without invoking uniqueness theorems or self-cited results as load-bearing. The generality assumption is an external hypothesis, not a self-definitional loop. Full text inspection (per instructions) yields no instances matching the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no information on free parameters, background axioms, or new entities; ledger left empty.

pith-pipeline@v0.9.1-grok · 5609 in / 1015 out tokens · 22401 ms · 2026-07-03T18:26:55.251094+00:00 · methodology

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Reference graph

Works this paper leans on

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