COMMENTS ON:

THE ALGEBRA AND GEOMETRY OF STEINER AND OTHER QUADRATICALLY PARAMETRIZABLE SURFACES

ADAM COFFMAN

Last update: Sept. 29, 2024.

1. Errata

The following typos appear in the published paper, [CSS].

• (p. 270) The derivative, equation (34), should read

\frac{∂f_{3}}{∂x} = [x_{2}(x_{3}^{2} + x_{4}^{2}) − 2x_{1}x_{4}^{2}, x_{1}(x_{3}^{2} + x_{4}^{2}),

2x_{1}x_{2}x_{3} − 4x_{3}(x_{3}^{2} + x_{4}^{2}), 2x_{1}x_{2}x_{4} − 2x_{1}^{2}x_{4} − 4(x_{3}^{2} + x_{4}^{2})x_{4}].

This corrects the lower right 2 × 2 block of the Hessian, equation (35), to

\begin{matrix}2x_{1}x_{2} − 12x_{3}^{2} − 4x_{4}^{2} & −8x_{3}x_{4} \\ −8x_{3}x_{4} & 2x_{1}x_{2} − 2x_{1}^{2} − 12x_{4}^{2} − 4x_{3}^{2} \end{matrix} ⌋ .

• (p. 273) Below equation (58) should read “As in the case of Σ_{7} and Σ_{8}”

• (p. 279) Theorem 7. should read “The order of Σ is 4 − ν_{Σ}.”

• (p. 281) Above equation (105) should read “Since P(Σ) ⊈ 𝕄_{2}, e ≠ 0.”

• (p. 284) Step (2) in Section 6. should read “det{λM + μN}.”

2. Updates

The references mention unpublished notes of A. Schwartz and C. Stanton — a version from 1988 is available from me, on request. Another pre-1996 source on this topic, with some details on the matrix calculations leading to the classification theorem, is [C_{1}], which I made available online in 2018.

It should have been mentioned that Coffman’s research was supported in part by a National Science Foundation Research Experience for Undergraduates program in the summer of 1990.

Since publication, the following related article has come to our attention: [D]. The topic of projections of the real Veronese variety has more recently been considered in [C_{3}]. Also see my web page on Steiner surfaces, currently at this address: [C_{2}].

3. Citations

Our article is cited in these academic papers: [A_{1}], [A_{2}], [A_{3}], [ABB], [AMT], [AS_{1}], [AS_{2}], [BJKL], [BOR], [BCF], [BEG], [CFRV], [EGL_{1}], [EGL_{2}], [GS], [G], [HJS], [H], [HW_{1}], [HK], [HW_{2}], [HW_{3}], [KO], [LG], [L], [M], [PA], [PL], [PO], [POS], [PR], [PT], [P], [RJ], [S], [SPS], [VMD], [WG], [WC], [WCD], [Y], [Zanella], [Z_{1}], [Z_{2}], [Z_{3}], as well as these books: [F], [KI], [OSG], and this computer technical manual: [T].

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Department of Mathematical Sciences, Purdue University Fort Wayne, Fort Wayne, IN 46805-1499

Email address: CoffmanA@pfw.edu