COMMENTS ON:

REAL CONGRUENCE OF COMPLEX MATRIX PENCILS AND COMPLEX PROJECTIONS OF REAL VERONESE VARIETIES

ADAM COFFMAN

Last update: July 7, 2021.

1. Errata

The following typos appear in the published paper, [C_{3}].

Line 3 of the Proof of Theorem 2.13: should read “As in the proof of” ….

The formula for τ in Example 8.5 at the top of page 82 should read:

τ : (z_{1}, iz_{1}, z_{3}, z_{4})^{T} ↦ (−(z_{1} + z_{3} + iz_{4}), −i(z_{1} + z_{3} + iz_{4}), z_{3}, z_{4})^{T}

2. Updates

In addition to reference [19] by Morley and Morley, there is another book which considers quadratically parametrized real curves in the complex plane, as in Section 4. [D] arrives at essentially the same classification as that in Section 4, and goes into some detail on the relationship between the complex coefficients and the geometry of the curve.

The Veronese surface and its projections, including Steiner surfaces (Examples 2.8, 7.4) are discussed in [B] and [SR]. These books also describe ([B] §§II, IV; [SR] §III.1) the connection between projective equivalence classes of rational parametric maps and linear systems of varieties, which is briefly mentioned in Example 7.6. A complete list of the 13 types of maps from Example 7.6 is given by Degtyarev (reference [13]), and also appears on my web site: [C_{4}].

Reference [5] will not appear under that title. However, due to a convenient editorial error, reference [5] is not cited in the paper anyway. The 4-manifold M in ℂ^{5} from Example 8.5, and other varieties defined by similar quadratic expressions, are considered in [C_{1}], [C_{2}], [C_{6}], and [C_{7}].

References [6] and [8] have appeared as cited below: [C_{5}], [CF].

A second edition of reference [24] has appeared as [SK], with a new list of authors. It still describes the vech notation but omits the congruence classification of real symmetric matrices.

My contact information has changed, and the web address at the end of the article is now obsolete. My current home page can be found at:

  http://users.pfw.edu/CoffmanA/

3. Citations

The article is cited in this paper: [W].

References

[B] A. Beauville, Complex Algebraic Surfaces, 2nd ed., LMSST 34, Cambridge, 1996. MR 1406314 (97e:14045), Zbl 1001.32010.

[C_{1}] A. Coffman, Enumeration and Normal Forms of Singularities in Cauchy-Riemann Struc-

tures, dissertation, University of Chicago, 1997. MR 2716702. [C_{2}] A. Coffman, Complexification of the CR cross-cap, unpublished lecture notes based on [C_{1}],

1997.

http://users.pfw.edu/CoffmanA/pdf/complexif.pdf

[C_{3}] A. Coffman, Real congruence of complex matrix pencils and complex projections of real Veronese varieties, Linear Algebra and its Applications, 370 (2003), 41–83. MR 1994320 (2004f:14026), Zbl 1049.14042.

[C_{4}] A. Coffman, Linear systems of conics, web page.

http://users.pfw.edu/CoffmanA/pov/lsoc.html

[C_{5}] A. Coffman, Analytic normal form for CR singular surfaces in ℂ^{3}, Houston J. Math (4) 30

(2004), 969–996. MR 2110245 (2006d:32048), Zbl 1074.32013. [C_{6}] A. Coffman, Analytic stability of the CR cross-cap, Pacific J. Math. (2) 226 (2006), 221–

258. MR 2247863 (2007j:32038), Zbl 1123.32018. [C_{7}] A. Coffman, Unfolding CR Singularities, Memoirs of the AMS (962) 205 (2010). MR

2650710 (2011f:32077), Zbl 1194.32016.

[CF] A. Coffman and M. Frantz, Möbius transformations and ellipses, The Pi Mu Epsilon Journal (6) 12 (2007), 339–345.

[D] R. Deaux, Introduction to the Geometry of Complex Numbers, transl. from the revised French ed. by H. Eves, Ungar Pub. Co., New York, 1957. MR 0086745 (19,236f), Zbl 0077.14502.

[SK] S. Searle and A. Khuri, Matrix Algebra Useful for Statistics, 2nd ed., Wiley, 2017. MR 3644547, Zbl 1365.62004.

[SR] J. G. Semple and L. Roth, Introduction to Algebraic Geometry, Oxford, 1985. MR 0814690 (86m:14001), Zbl 0576.14001.

[W] W. Waterhouse, Real congruences of complex subspaces of 2 × 2 symmetric complex matrices, Linear Algebra and its Applications (2–3) 414 (2006), 502–505. MR 2214402 (2006j:15038), Zbl 1091.15033.

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

E-mail address: CoffmanA@pfw.edu