Generalization of Pascal's theorem to higher dimensions

Question

Pascal's celebrated theorem in classical geometry gives a necessary and sufficient condition for the existence of a conic through six given points in the plane. Does there exists a similar statement which gives a geometric condition for the existence of a quadric through $\frac{(N+1)(N+2)}{2}$ points in the $N$-dimensional projective space?

Answer 1

The question is not totally clear since an obvious necessary and sufficient condition is the nonvanishing of the multivariate Vandermonde determinant $V$, namely the $M\times M$ matrix with $M=(N+1)(N+2)/2$, made of all homogeneous coordinate monomials of degree 2 for these $M$ points. I guess what is asked for is a reasonably natural geometric condition like saying that three points built out of the original six are aligned. This is a difficult question even for $N=3$, studied by Turnbull and Young. It is related to the question in my comment to Will Sawin's answer to this MO question. As an $SL_{N+1}$ invariant, the multivariate Vandermonde $V$ has an expression in terms of $(N+1)\times(N+1)$ determinants of the point coordinates. This follows from the Cayley-Clebsch Theorem about the existence of symbolic formulas for classical invariants. This is nowadays called the "first fundamental theorem of classical invariant theory for $SL_{N+1}$". Because of the Plücker relations these expressions are highly nonunique.

Pascal's theorem is about suitable explicit such formulas. See in particular formula (3) on page 450 in the article "On the synthetic factorization of projectively invariant polynomials" by Sturmfels and Whiteley. This formula is probably due to Olry Terquem and appeared in the Nouvelles Annales de Mathématiques of 1857. See this link.

For the case $N=2$, let $a=(a_1,a_2,a_3)$, $b=(b_1,b_2,b_3)$ etc. denote the homogeneous coordinates of the six points $A,B,C,D,E,F$. Then $$ V=\left| \begin{array}{cccccc} a_1^2 & a_1 a_2 & a_1 a_3 & a_2^2 & a_2 a_3 & a_3^2 \\ b_1^2 & b_1 b_2 & b_1 b_3 & b_2^2 & b_2 b_3 & b_3^2 \\ c_1^2 & c_1 c_2 & c_1 c_3 & c_2^2 & c_2 c_3 & c_3^2 \\ d_1^2 & d_1 d_2 & d_1 d_3 & d_2^2 & d_2 d_3 & d_3^2 \\ e_1^2 & e_1 e_2 & e_1 e_3 & e_2^2 & e_2 e_3 & e_3^2 \\ f_1^2 & f_1 f_2 & f_1 f_3 & f_2^2 & f_2 f_3 & f_3^2 \end{array} \right| $$ and the symbolic formula I am talking about is $$ V=-(abc)(aef)(bfd)(cde)+(def)(dbc)(eca)(fab) $$ where the bracket factors like $(abc)$ denote point coordinate determinants: $$ (abc)=\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right| $$ etc. As explained on page 269 of the article "The linear invariants of ten quaternary quadrics" by Turnbull and Young, the coordinates of the three points which should be aligned are given by $$ \begin{array}{ccc} \alpha_i & = & (dcf)e_i-(dce)f_i \\ \beta_i & = & (cae)b_i -(cab)e_i \\ \gamma_i & = & (afb)d_i -(afd)b_i \end{array} $$ for $1\le i\le 3$. They are aligned iff $(\alpha \beta \gamma)=0$ which is the same as requiring $$ -(abc)(aef)(bfd)(cde)+(def)(dbc)(eca)(fab)=0 $$

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Addendum: formulas like the one above are useful for constructing catalecticants or equations for secants of Veronese varieties, see the last section of this preprint. One of the first references along this line of thought is Clebsch's article on quartic curves, which is where I found the citation to the above 1857 NAM article.

Answer 2

See the very nice paper by Eisenbud, Green, and Harris:

@article {MR1376653,
    AUTHOR = {Eisenbud, David and Green, Mark and Harris, Joe},
     TITLE = {Cayley-{B}acharach theorems and conjectures},
   JOURNAL = {Bull. Amer. Math. Soc. (N.S.)},
  FJOURNAL = {American Mathematical Society. Bulletin. New Series},
    VOLUME = {33},
      YEAR = {1996},
    NUMBER = {3},
     PAGES = {295--324},
      ISSN = {0273-0979},
     CODEN = {BAMOAD},
   MRCLASS = {14N05 (14M05)},
  MRNUMBER = {1376653 (97a:14059)},
MRREVIEWER = {Raquel Mallavibarrena},
       DOI = {10.1090/S0273-0979-96-00666-0},
       URL = {http://dx.doi.org/10.1090/S0273-0979-96-00666-0},
}