arxiv: 2605.09437 · v1 · submitted 2026-05-10 · 🧮 math.AG

Recognition: 2 theorem links

· Lean Theorem

On the tangent degree and the degree of the tangent variety of a projective variety

Francesco Russo, Jordi Hernandez Gomez

Pith reviewed 2026-05-12 02:06 UTC · model grok-4.3

classification 🧮 math.AG

keywords tangent degreetangent varietyprojective varietysecant varietydegree boundsclassificationalgebraic geometry

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The pith

The tangent degree of a projective variety is never one when ambient dimension equals twice the variety dimension, and the tangent variety degree has a linear lower bound in its codimension when distinct from the secant variety.

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

The paper defines the tangent degree τ(X) as the number of tangent spaces to a projective variety X through a general point of its tangent variety Tan(X), or zero when that dimension is too low. It proves that τ(X) cannot equal 1 when the ambient projective space has dimension exactly twice the dimension of X. When Tan(X) differs from the secant variety, the paper derives a linear lower bound on the degree of Tan(X) in terms of the codimension of Tan(X). It then locates the varieties attaining these bounds in low dimensions or under smoothness, and classifies those with τ(X) greater than one for higher ambient dimensions in small cases.

Core claim

For a projective variety X^n subset P^N, the tangent degree τ(X) is never equal to 1 when N equals 2n. When Tan(X) does not coincide with the secant variety, deg(Tan(X)) satisfies a linear lower bound in terms of the codimension of Tan(X) in P^N. These invariants reach their lower bounds only in specific low-dimensional or smooth cases, while varieties with τ(X) > 1 are classified when N is at least 2n+1 and dimensions are small.

What carries the argument

The tangent degree τ(X), counting tangent spaces through a general point of Tan(X) when positive and finite, together with the degree of the tangent variety Tan(X) itself.

Load-bearing premise

The tangent variety must differ from the secant variety for the linear lower bound on its degree to apply.

What would settle it

A counterexample would be a variety X^n in P^{2n} where τ(X) equals 1, or any variety where Tan(X) differs from the secant variety yet deg(Tan(X)) falls below the stated linear bound in codimension.

read the original abstract

The tangent degree $\tau(X)$ of a projective variety $X^n\subset\mathbb P^N$ is the number of tangent spaces to $X$ at smooth points passing through a general point of the tangent variety $Tan(X)\subseteq\mathbb P^N$, if positive and finite; it is equal to zero if $\dim(Tan(X))<2n$. In this paper we focus on general properties of $\tau(X)$ and of $deg(Tan(X))$. For example $\tau(X)\neq 1$ if $N=2n$ and, as soon as $Tan(X)$ does not coincide with the secant variety, we prove a linear lower bound for the degree of $Tan(X)$ in terms of its codimension in the spirit of the paper Ciliberto.Russo.2006. Then we consider the cases in which the previous two invariants attain the lower bounds found here, either in small dimension/codimension and/or under the smoothness assumption. Finally for $N\geq 2n+1$ we consider varieties $X^n\subset\mathbb P^N$ having $\tau(X)>1$ and provide their classification in small dimension.

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.

Desk Editor's Note private letter to a colleague

The paper proves τ(X) ≠ 1 when N=2n and a linear lower bound on deg(Tan(X)) when it differs from the secant variety, plus low-dimensional classifications, but only the abstract is available so the proofs remain unchecked.

read the letter

The two things to know right away are that this paper proves the tangent degree τ(X) is never 1 when the ambient dimension N equals twice the variety dimension n, and that it establishes a linear lower bound for the degree of the tangent variety in terms of its codimension whenever Tan(X) does not coincide with the secant variety. Both are framed as extensions of the Ciliberto-Russo 2006 results, followed by classifications of cases that attain these bounds in small dimensions or under smoothness assumptions, and some work on varieties with τ(X) > 1 when N is at least 2n+1. What the paper does well is supply these concrete properties and then narrow in on explicit low-dimensional lists, which can be useful for people who actually compute degrees or classify projective varieties. The approach stays within standard generality assumptions on points and tangent spaces, so it feels like a direct, incremental follow-up rather than a big departure. The soft spots are real but limited by what we can see. Only the abstract is available, so there is no way to verify whether the arguments are complete, whether they handle the boundary cases cleanly when dimensions are close, or if any hidden gaps appear in the proofs of the non-equality or the bound. The classification work is restricted to small dimensions, which is reasonable but leaves open how far the results extend. No circularity or invented notions show up in the given summary. This is aimed at algebraic geometers who already work with tangent and secant varieties of projective varieties and know the 2006 paper. A reader in that niche would get practical value from the new inequalities and the explicit cases. It deserves a serious referee to check the details once the full text is in hand.

Referee Report

0 major / 1 minor

Summary. The paper defines the tangent degree τ(X) of a projective variety X^n ⊂ ℙ^N as the number of tangent spaces at smooth points of X passing through a general point of the tangent variety Tan(X), or zero if dim(Tan(X)) < 2n. It establishes general properties of τ(X) and deg(Tan(X)), proving in particular that τ(X) ≠ 1 whenever N = 2n, and that deg(Tan(X)) satisfies a linear lower bound in terms of codim(Tan(X)) whenever Tan(X) is distinct from the secant variety (extending Ciliberto-Russo 2006). The manuscript further studies extremal cases attaining these bounds in low dimension or codimension (including under smoothness assumptions) and classifies varieties with τ(X) > 1 for N ≥ 2n+1 in small dimensions.

Significance. If the stated results and proofs hold, the work provides useful extensions of known bounds on tangent varieties and their degrees, together with a classification in low dimensions. These could aid further study of defective varieties and relations between tangent and secant varieties in algebraic geometry. The abstract indicates the arguments rely on standard generality assumptions on points and tangent spaces, with no free parameters or ad-hoc axioms apparent.

minor comments (1)

The abstract refers to results 'in the spirit of' Ciliberto-Russo 2006 without specifying the precise relation or differences; a short comparison paragraph in the introduction would clarify the novelty.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting its potential utility in extending bounds on tangent varieties. The recommendation is listed as uncertain, but the report provides no specific major comments after the heading 'MAJOR COMMENTS:'. Accordingly, we have no individual referee points to address or revise at this stage. We remain confident in the results, which rely on standard generality assumptions as indicated in the abstract.

Circularity Check

0 steps flagged

No significant circularity; results presented as independent proofs

full rationale

The abstract defines τ(X) explicitly and states that the paper proves τ(X) ≠ 1 when N = 2n and a linear lower bound on deg(Tan(X)) (when Tan(X) does not coincide with the secant variety) 'in the spirit of' the 2006 Ciliberto-Russo reference. These are framed as new results proved here under standard generality assumptions, with subsequent classification of equality cases also claimed as original. No equation or claim reduces by construction to a fitted input, self-definition, or load-bearing self-citation; the 2006 reference supplies methodological analogy rather than the target statements themselves. The derivation chain therefore remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard definitions and axioms from algebraic geometry rather than new free parameters or invented entities. The lower bounds are proved in the spirit of prior work.

axioms (2)

standard math Projective varieties are defined over an algebraically closed field with tangent spaces at smooth points.
This is standard in algebraic geometry and invoked in the definition of τ(X) and Tan(X).
domain assumption The tangent variety Tan(X) is the closure of the union of tangent spaces at smooth points.
Used throughout the definitions and properties studied in the paper.

pith-pipeline@v0.9.0 · 5475 in / 1621 out tokens · 86083 ms · 2026-05-12T02:06:41.859254+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
The tangent degree τ(X) of a projective variety X^n ⊂ ℙ^N is the number of tangent spaces to X at smooth points passing through a general point of the tangent variety Tan(X) ⊆ ℙ^N
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
we prove a linear lower bound for the degree of Tan(X) in terms of its codimension

Reference graph

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