# Algebraic Geometry Formulae

Antreas P. Hatzipolakis xpolakis at otenet.gr
Fri Sep 21 20:05:37 CEST 2001

```Dear Neil,

I am copying the following formuale from the paper:

Luigi Berzolari: Allgemeine Theorie der Ho"heren Ebenen Algebraischen Kurven.
Encyclopa"die der Mathematischen Wissenschaften mit Einschluss ihrer
Anwendungen. Band III_2. Heft 3.
Leipzig, Druck und Verlag von B.G. Teubner, 1906. SS. 313 - 455.

In p. 341:

Hesse'sche Kurve | Steiner'sche Kurve  | Cayley'sche Kurve
__|__________________________________________________________________________
|
N |3(n-2)       | 3(n-2)^2     |    3(n-2)(5n-11)
|
N'|3(n-2)(3n-7) | 3(n-1)(n-2)  |    3(n-1)(n-2)
|
D |0  | 3(n-2)(n-3)(3n^2-9)(3n^2-9n-5)/2 | 9(n-2)(5n-13)(5n^2-19n+16)/2
|
D'|27(n-1)(n-2)(n-3)(3n-8)/2 | 3(n-2)(n-3)(3n^2-3n-8)/2 | 9(n-2)^2(n^2-2n-1)/2
|
R |0            |  12(n-2)(n-3) | 18(n-2)(2n-5)
|
R'|9(n-2)(3n-8) | 3(n-2)(4n-9)  |    0

-------------

In p. 352:

n(n+1)(n+2)/3

n(n+1)(n+2)(2n^3 + 6n^2 + 7n -3)/36

n(n^2 - 1)(n+2)(2n^5 + 14n^4 + 49n^3 + 91n^2 + 90n + 18)/324

-------------

In p. 353:

n(n+1)(n+2)/6

n(n^2 - 1)(n + 2)(n^2 + 4n + 6)/72

280C(n+4,9) + 280C(n+4,8) + 105C(n+3,7) + 77C(n+3,6) + 43C(n+2,5) -
- 16(n+2,4) + 20C(n+1,3) - [n(n^2 - 1)(n^2 - 4)(n-3)/360]

Sorry, I have no time to translate (with the help of dictionary, since
I don't speak German) the German text explaining the meaning
of the formulae.

Greetings from Athens

Antreas

PS: There are no typos in my typing of the formulae.
(I corrected them before sending the message!)

```