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Interpolation for degree 2 Veroneses of odd dimension

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read For odd n, at least $2^{n(n-1)}$ degree-2 Veronese varieties pass through any general points.

desk verdict Genuinely new construction and a likely-true theorem, but the proof skips a scheme-theoretic step that is load-bearing. read the letter →

arxiv 2411.16672 v1 pith:G4REJW36 submitted 2024-11-25 math.AG

Ray Shang This is my paper · ORCID
classification math.AG MSC 14N0514C2014H10
keywords interpolationdegree2VeronesevarietiesnormalbundlesrationalcurvechainsauxiliarycurvessquarerootsoflineflagHilbertschemesenumerativelowerbounds
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves interpolation for degree 2 Veronese varieties of odd dimension: for any positive odd $n$, through any $\binom{n+2}{2}+n+1$ general points in $\mathbb{P}^{\binom{n+2}{2}-1}_{\mathbb{C}}$ there pass at least $2^{n(n-1)}$ such varieties. These varieties are the images of $\mathbb{P}^n$ under the complete linear system $|\mathcal{O}_{\mathbb{P}^n}(2)|$. The argument reduces interpolation to the vanishing of normal vector fields, then exhibits a smooth auxiliary curve forcing that vanishing, and the count $2^{n(n-1)}$ is exactly the number of square roots of the restriction of $\mathcal{O}(1)$ to that curve. This settles the odd-dimensional case of the open interpolation question for degree 2 Veroneses and extends a classical surface computation to all odd dimensions.

What carries the argument

The load-bearing mechanism is the auxiliary curve obtained by smoothing a rational normal curve chain. The chain is a nodal curve built from $(n+1)/2$ rational normal curves in $\mathbb{P}^n$, and after smoothing it has degree $n(n+1)/2$ and genus $n(n-1)/2$. Its role is to convert the normal-bundle interpolation condition into a statement about square roots: the Veronese varieties containing the curve correspond bijectively to the square roots $L$ of $\mathcal{O}_{\tilde{C}}(1)$, and the flag Hilbert scheme argument turns this bijection into the vanishing $H^0(V,N_{V/\mathbb{P}}\otimes I_{\tilde{C}})=0$. Since a smooth curve of genus $n(n-1)/2$ has exactly $2^{n(n-1)}$ square roots, the correspondence simultaneously yields the enumerative lower bound.

What would settle it

For a small odd $n$ such as $n=3$, construct the smoothed auxiliary curve $\tilde{C}$ and compute the Zariski tangent space of the flag Hilbert scheme $\mathrm{Hilb}^{\tilde{C}}_{V}$ at the point $([\tilde{C}],[V])$, equivalently $H^0(V,N_{V/\mathbb{P}}\otimes I_{\tilde{C}})$; if this space is nonzero while the set-theoretic bijection holds, then Proposition 3.5 is not scheme-theoretic and the final vanishing used in the interpolation proof fails.

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Extended reading notes

Core claim

The central claim is that a rational normal curve chain degenerates, after smoothing, into an auxiliary curve that controls every Veronese through it. For odd $n$, take $(n+1)/2$ rational normal curves in $\mathbb{P}^n$, gluing consecutive curves nodally at $n+1$ general points each; the resulting chain has degree $n(n+1)/2$ and arithmetic genus $n(n-1)/2$. After the degree-2 Veronese embedding, sections of the Veronese normal bundle twisted by the prescribed marked points vanish on the chain, because the point insertions on the first rational component force its sections to zero and compatibility across the nodes forces the rest to zero. Smoothing preserves this vanishing by upper semicontinuity and produces a smooth curve $\tilde{C}$ with $\mathcal{O}_{\tilde{C}}(2)$ non-special and all its square roots very ample. For this curve, degree-2 Veronese varieties containing it are in bijection with square roots of $\mathcal{O}_{\tilde{C}}(1)$, so their number is $2^{2g}=2^{n(n-1)}$; the paper then uses lower semicontinuity of fiber components to carry this count to general point configurations.

Load-bearing premise

The load-bearing premise is that the correspondence between Veronese varieties containing the smoothed auxiliary curve and square roots of its restricted line bundle is an isomorphism of schemes, so that it has no hidden infinitesimal directions; the paper asserts this 'by the same argument' as the surface case without proving it, and a mere matching of points would not rule those directions out.

Editorial extensions

If this is right

  • Interpolation holds for degree 2 Veronese varieties of every odd dimension $n$: with the expected number $\binom{n+2}{2}+n+1$ of points, a general configuration always admits such a variety.
  • The number of such varieties through a general configuration is at least $2^{n(n-1)}$; for instance $n=3$ gives at least $64$ degree-2 Veronese threefolds through 14 points in $\mathbb{P}^9$.
  • Interpolation is exact rather than merely dominant: over a general point configuration the incidence fiber is zero-dimensional.
  • The rational-normal-curve-chain technique gives a template for proving interpolation of higher-dimensional varieties by degenerating to curves with known normal-bundle splitting and then smoothing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the scheme-theoretic bijection in Proposition 3.5 is verified explicitly, the same degeneration strategy is the natural route to even dimensions; the paper notes that rational normal curve chains have the wrong degree and genus there, so a different auxiliary curve class would be needed.
  • The count $2^{n(n-1)}$ suggests a general interpolation principle: whenever a smooth auxiliary curve of genus $g$ controls a family of varieties, the expected number of members through general points should be at least the number of theta characteristics $2^{2g}$; this is an inference, not a claim of the paper.
  • A direct computational test is available: for small odd $n$, smooth the chain, build the flag Hilbert scheme near the auxiliary curve, and check whether its tangent space is zero, thereby confirming or refuting the scheme-theoretic step that the paper leaves to 'the same argument'.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proves Theorem 1.1: for any positive odd integer n, there exist at least 2^{n(n-1)} degree 2 Veronese varieties of dimension n through any binom(n+2,2)+n+1 general points in P^{binom(n+2,2)-1}. The proof reduces interpolation for a Veronese variety V to the vanishing of H^0(V,N_{V/P^N}⊗I_C) for a well-chosen auxiliary curve C, constructs a rational normal curve chain in P^n, verifies the required vanishing on this chain, smooths the chain to a smooth curve C~ whose hyperplane bundles and their square roots are non-special and very ample, and then invokes Proposition 3.5 to pass from a bijection with square roots of O_{C~}(1) to the vanishing H^0=0 and to the enumerative lower bound.

Significance. If correct, the paper gives a substantial new result in higher-dimensional interpolation, extending Coble's degree-2 Veronese surface theorem and providing progress on a question of Landesman and Patel. The use of normal bundle restrictions and rational normal curve chains is a promising technique, and the explicit computations for the chain (Propositions 4.4 and 4.5) are coherent and clearly presented. However, the proof has a load-bearing gap: the scheme-theoretic promotion of the bijection in Proposition 3.5 is asserted without proof, and a key splitting type is quoted from an unpublished preprint [Sha24]. The enumerative lower bound also relies on a semicontinuity statement that appears to be misstated. These issues need to be addressed before the result can be considered established.

major comments (3)
  1. [Section 3.3, after Prop. 3.5; Eq. (4)] The proof requires that H^0(V,N_{V/P^N}⊗I_C)=0, which is the tangent space of the flag Hilbert scheme component Hilb_C^V at ([C],[V]). The paper asserts 'by the same argument discussed in Subsection 3.2' that there is a scheme-theoretic isomorphism between Hilb_C^V and a translate of Pic(C)[2], but this assertion is not proved. Proposition 3.5 establishes only a bijection of closed points between degree 2 Veronese varieties containing C and square roots of O_C(1). A set-theoretic bijection with a reduced finite scheme does not imply that the flag Hilbert scheme is reduced; for example, Spec k[ε]/(ε^2) → Spec k is bijective on closed points but has a nonzero tangent vector. The surface-case argument in Subsection 3.2 uses connectedness of Hilb_C^V and normality of the target, neither of which is established for higher-dimensional C. This gap is load-bearing: a nonzero section of H^0(V,N⊗I_C) would be a first-order deformation of V inside P^N with C fixed, and would break the reduction to interpolation. Please provide a proof of the scheme-theoretic isomorphism or an alternative argument establishing reducedness of Hilb_C^V at the relevant point.
  2. [Proposition 4.6] The splitting type N_{V_{n,2}/P^N}|_{R_i} ≅ ⊕ O_{P^1}(2n+2) is quoted from the author's own preprint [Sha24, Theorem 4.3] without proof. This splitting type is the starting point of the vanishing argument for the rational normal curve chain, which is then propagated by upper semicontinuity to the smoothed auxiliary curve. Since [Sha24] is not published and the result is not proved in the present paper, the proof is not self-contained at a load-bearing step. Please include a proof of the splitting type or provide a publicly available reference with a complete proof.
  3. [Section 5, final paragraph] The statement 'By [DM69, Theorem 4.17(iii)], the number of connected components of the geometric fibers of this projection map is a lower semicontinuous function' appears to reverse the usual semicontinuity for proper morphisms; the cited theorem (if it is the standard one) gives upper semicontinuity. If so, the conclusion that there is a dense open subset over which the fibers have at least 2^{n(n-1)} connected components does not follow as written. The desired lower bound can instead be obtained from the constancy of the degree of the generically finite projection over a dense open (or from generic flatness), but the text needs to be corrected and justified. Since the lower bound is part of Theorem 1.1, this point should be fixed.
minor comments (4)
  1. [Section 4, first line] The ambient projective space in the degree 2 Veronese embedding is written as P^{binom(n+2,2)}, but the notation in Section 1.2 and elsewhere uses P^{binom(n+2,2)-1}; the exponent is off by one.
  2. [Section 3.5, proof of Prop. 3.5, backward direction] The backward direction says 'we obtain a degree 2 Veronese variety which contains C' without explicitly explaining how a projective automorphism is used to make the constructed Veronese contain the original curve C rather than a projectively equivalent copy. Please clarify this step.
  3. [Section 5, final paragraph] The projection map in the incidence correspondence is written with a product of Grassmannians Gr(1, ...), but for point interpolation the base should be the product of points in P^N; the notation conflates the general λ-interpolation setup with the point-interpolation specialization. Please clarify which map is being discussed.
  4. [Section 5, proof of Prop. 5.1] The notation T^C_1, T^C_2, T^C, T^D, T^final is heavy and somewhat confusing; consider using a single open subset obtained by successive intersections.

Circularity Check

1 steps flagged · score 4.0 of 10

One load-bearing splitting-type input is imported from the author's own unpublished preprint; the rest of the derivation is self-contained, though the scheme-theoretic isomorphism step is an unproved gap rather than circular.

  1. self citation load bearing [Section 4, proof of Proposition 4.6; cf. Remark 4.7]
    "N_{V_{n,2}/P^{\binom{n+2}{2}-1}_{\mathbb C}}|_{R_i} \cong \bigoplus_{i=1}^{n(n+1)/2} O_{\mathbb P^1}(2n+2) by [Sha24, Theorem 4.3]."

    The splitting type of the Veronese normal bundle along each rational normal curve is the key numerical input for Proposition 4.6, which produces the vanishing H^0(C, N_{V/P^N}|_C \otimes I_{points})=0 on the degenerate chain. That vanishing is then propagated by upper semi-continuity to the smooth auxiliary curve and is ultimately used to prove Theorem 1.1. The only justification given in this paper is a citation to the author's own preprint [Sha24], whose theorem is not stated, proved, or independently verified here. Thus a load-bearing step of the derivation reduces to a self-citation rather than to a proof contained in or externally checked for this paper.

full rationale

The main numerical content—the count of square roots, the construction and smoothing of the rational normal curve chain, and the upper semi-continuity argument—is self-contained and does not assume Theorem 1.1. The reduction to normal-bundle interpolation in Proposition 3.1 is a standard equivalence, and Proposition 3.5 gives a bijection whose use to count Veroneses is legitimate. However, one load-bearing input is not independently established in the paper: Proposition 4.6's splitting type is cited to the author's own unpublished preprint [Sha24, Theorem 4.3], and without that splitting the vanishing on the chain, and hence the smoothed auxiliary-curve condition (2), does not follow. This raises the circularity score above the 0-2 range. Separately, the paper's use of a scheme-theoretic isomorphism between Hilb_C^{V_{n,2}} and a translate of Pic(C)[2] (Section 3.3, used at equation (4)) is asserted 'by the same argument' and not proved; a bijection of closed points alone would not rule out nilpotent tangent directions. That is a proof gap affecting correctness, not a circular reduction of the conclusion to the hypotheses. The central theorem itself is not assumed in the inputs, so the circularity is partial and localized rather than total.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central proof rests on a small number of auxiliary objects and cited results. The new construction (rational normal curve chain) is well-defined, but two key inputs are the author's own prior splitting theorem and an unproved scheme-theoretic version of the Coble bijection.

assumptions (4)
  • domain assumption Normal bundle of the degree 2 Veronese along a rational normal curve splits as O(2n+2)^(n(n+1)/2) (Theorem 4.3 of [Sha24]).
    Used in Proposition 4.6 to obtain vanishing sections on the rational normal curve chain; cited from the author's previous preprint without proof in this paper.
  • ad hoc to paper The bijection of Proposition 3.5 extends to a scheme-theoretic isomorphism between the flag Hilbert scheme Hilb^C_V and a translate of Pic(C)[2].
    Needed to conclude H^0(V,N⊗I_C)=0; the paper does not prove this scheme-theoretic statement and set-theoretic bijection alone is insufficient.
  • domain assumption The Hilbert scheme of linearly normal curves of degree n(n+1)/2 and genus n(n-1)/2 in P^n is irreducible ([Kee22]).
    Used in Proposition 5.1 to deform the auxiliary curve to one with desired square-root properties.
  • standard math Standard deformation theory: a nodal curve with H^1(C,O_C(1))=0 smooths as an embedded curve (via [Har10, Prop 29.9] and the Euler sequence).
    Used to construct the family X over B; the implication from H^1(O(1)) to H^1(N) is not spelled out.
invented entities (1)
  • rational normal curve chain
    purpose: A nodal degenerate curve used to construct smooth auxiliary curves via smoothing; it is a chain of rational normal curves glued at n+1 points between consecutive components.
    This is a construction introduced in this paper; it has no independent empirical handle and exists only as a mathematical object.

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Pith. "Pith review of Interpolation for degree 2 Veroneses of odd dimension." pith.science (2026). https://pith.science/paper/G4REJW36

@misc{pith2026241116672,
  author       = {Pith},
  title        = {Pith review of: Interpolation for degree 2 Veroneses of odd dimension},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G4REJW36}},
  note         = {Machine review of arXiv:2411.16672}
}
abstract

A classical fact is that through any $d+3$ general points in $\mathbb{P}_\mathbb{C}^d$ there exists a unique rational normal curve of degree $d$ passing through them. We generalize this by proving the following: when $n$ is odd, for any $\binom{n+2}{2} + n+1$ general points in $\mathbb{P}_\mathbb{C}^{\binom{n+2}{2} - 1}$, there exist at least $2^{n(n-1)}$ degree 2 Veroneses passing through them. This makes substantial progress on a question of Aaron Landesman and Anand Patel, and extends the work of Arthur Coble.

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