Section 2: Algebra and CombinatoricsΒΆ

The eight coefficients of the octanomial model

\[a \cdot xyz + b \cdot xyw + c \cdot xzw + d \cdot yzw + e \cdot x^2 y + f \cdot x y^2 + g \cdot z^2 w + h \cdot z w^2\]

are quintics in the moduli parameters \(d_1, d_2,d_3,d_4, d_5, d_6\). They were found by a calculation in Macaulay2 carried out with the help of Mike Stillman and described in the proof of Proposition 2.1.

\[\begin{split}\begin{matrix} a \,= & d_1 d_3 d_2 d_4 ( d_1{+} d_3{-} d_2{-} d_4)+ d_2 d_4 d_5 d_6 ( d_2{+} d_4{-} d_5{-} d_6)+ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \\ & d_5 d_6 d_1 d_3 ( d_5{+}d_6{-}d_1{-}d_3) + d_5 d_6 ( d_5{+} d_6) ( d_1^2{+}d_3^2{-}d_2^2{-}d_4^2)+ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad\\ & d_1 d_3 ( d_1{+} d_3) ( d_2^2{+} d_4^2{-} d_5^2{-} d_6^2) + d_2 d_4 ( d_2{+} d_4) ( d_5^2{+} d_6^2{-} d_1^2{-} d_3^2) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ b \,= & d_1 d_2 d_3 d_5 ( d_1{+} d_2{-} d_3{-} d_5)+ d_3 d_5 d_4 d_6 ( d_3{+} d_5{-} d_4{-} d_6)+ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \\ & d_4 d_6 d_1 d_2 ( d_4{+} d_6{-} d_1{-} d_2) + d_4 d_6 ( d_4{+} d_6) ( d_1^2{+} d_2^2{-} d_3^2{-} d_5^2)+ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \\ & d_1 d_2 ( d_1{+} d_2)( d_3^2{+} d_5^2{-} d_4^2{-} d_6^2)+ d_3 d_5 ( d_3{+} d_5) (d_4^2{+} d_6^2{-} d_1^2{-} d_2^2) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ c \, = & d_1 d_3 d_2 d_5 ( d_1{+} d_3{-} d_2{-} d_5)+ d_2 d_5 d_4 d_6 ( d_2{+} d_5{-} d_4{-} d_6)+ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \\ & d_4 d_6 d_1 d_3 ( d_4{+} d_6{-} d_1{-} d_3) + d_4 d_6 ( d_4{+} d_6) ( d_1^2{+} d_3^2{-} d_2^2{-} d_5^2)+ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \\ & d_1 d_3 ( d_1{+} d_3) ( d_2^2{+} d_5^2 {-} d_4^2{-} d_6^2)+ d_2 d_5 ( d_2{+} d_5) ( d_4^2{+} d_6^2{-} d_1^2{-} d_3^2) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ d \,= & d_1 d_2 d_3 d_4 ( d_1{+} d_2{-} d_3{-} d_4)+ d_3 d_4 d_5 d_6 ( d_3{+} d_4{-} d_5{-} d_6)+ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \\ & d_5 d_6 d_1 d_2 ( d_5{+} d_6{-} d_1{-} d_2) + d_5 d_6 ( d_5{+} d_6) ( d_1^2{+} d_2^2{-} d_3^2{-} d_4^2)+ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \\ & d_1 d_2 ( d_1{+} d_2) ( d_3^2{+} d_4^2{-} d_5^2{-} d_6^2) + d_3 d_4 ( d_3{+} d_4) ( d_5^2{+} d_6^2{-} d_1^2{-} d_2^2) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ e \,= & -( d_1+ d_3+ d_5) ( d_2+ d_4+ d_6) ( d_1- d_5) ( d_2- d_6) ( d_3- d_4) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ f \,= & -( d_1+ d_2+ d_4) ( d_3+ d_5+ d_6) ( d_1- d_4) ( d_2- d_5) ( d_3- d_6) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ g \, = & -( d_1+ d_3+ d_4) ( d_2+ d_5+ d_6) ( d_1- d_4) ( d_2- d_6) ( d_3- d_5) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \\ h \, = & -( d_1+ d_2+ d_5) ( d_3+ d_4+ d_6) ( d_1- d_5) ( d_3- d_6) ( d_2- d_4) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \end{matrix}\end{split}\]

The file all_lines contains the formulas for all \(27\) lines and the coordinates of their \(135\) intersection points in term of the moduli parameters. See the file all_pointspols for the \(135\) intersection points with polynomial coordinates.

The file matrix contains a \(3\times 3\) matrix whose entries are linear forms in \(x,y,z,w\) and coefficients in \(d_1, d_2,d_3,d_4, d_5, d_6\). Its determinat gives the determinantal representation of our octanomial model.

The statement in Remark 2.2 can be verified using the following maple code.

The support of the octanomial surface is

\[\mathcal{A} \, = \, \bigl\{ (1110), (1101), (1011), (0111), (2100), (1200), (0021), (0012) \bigr\}.\]

The list of the \(53\) unimodular triangulations of this point configuration can be found here.