Aggressive Geometry – The degree of hypersurface of pfaffian cubic quadruplets

Let $ Pi: = mathbb {P} (H ^ 0 ( mathbb {P} ^ 5, mathcal {O} _ { mathbb {P}} (3))) $ to be four cubic space $ mathbb {P} ^ $ 5. It is well known that these cubic ones that are Pfaffian, that is, defined by the pfaffian of asymmetric symmetry of 6 out of 6
matrix of linear forms, forms a hypersurface $ Pi $. Does anyone know the degree of this hypersurface?

nt.number theory – How did Gauss find the units of the cubic body $ Q[n^{1/3}]$?

I recently read an article on jstor "Gauss and the early development of algebraic numbers", which describes the genesis of Gauss's ideas on the foundations of the algebraic theory of numbers, among other useful information, he mentions a certain ternary cubic form that Gauss studied in 1808 to try to understand the principles under higher rates of reciprocity (cubic reciprocity in this case).

The particular form is:
$$ F (x, y, z) = x ^ 3 + ny ^ 3 + n ^ 2z ^ 3 – 3nxyz $$ and Gauss attempted to find (rational) solutions to the Diophantine equation $ F (x, y, z) = $ 1. As the article explains, this particular form appears as the norm of the number $ x + vy + v ^ 2z $ (or $ v = n ^ {1/3} $) in the pure cubic field created by joining $ v $ the field of rationals. Since Gauss wanted to know where this expression was equal to 1, this investigation can be interpreted as an attempt to find the units (norm 1 numbers) in this cubic field. Gauss then recorded the units for some values ​​of n and, in some cases, presented the fundamental unit.

I have not found enough information about this Gaussian investigation. So now, to my questions:

  • What was the Gauss procedure? And how does this relate to Gauss's other investigations in the algebraic theory of numbers?

  • Does this have anything to do with the Dirichlet Unity Theorem?? I ask the question because this article says that the Gaussian investigation was "a step in the progression of Lagrange to Dirichlet, the latter having developed in 1842-1846 the general theory of algebraic units …".

Cubic numbers in C ++

How can I find the cubic numbers in C ++, adding only, without
use multiplications as such, "*". It must go into an arrangement
from 15.

void Cubicos () {

for (int i = 1; i <= 15; i ++) {
B[i] = pow (i, 3);

Algebraic geometry – Cubic surface with a singularity A5

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Agalic geometry – Cubic surfaces and 6-point configurations

A smooth and cubic surface $ X subset mathbb {P} ^ $ 3 is isomorphic to $ mathbb {P} ^ 2 $ exploded at six points, so there should be a birational map

$ H ^ 0 ( mathbb {P} ^ 3, mathscr {O} _ { mathbb {P} ^ 3} (3)) // PGL_4 dashrightarrow { rm Hilb} ^ 6 mathbb {P} ^ $ 2

Given a family with 1 parameter $ X_t $ smooth cubic surfaces specialized to $ X_0 $how are the six points degenerating? For example, I know that $ X_0 $ have a $ A_1 $ the singularity could be 3 points becoming collinear or 6 points located on a conic (contributing to the collapse of a curve (-2) under the canonical map), but I do not know references for other singularities.