Plücker coordinates: Difference between revisions

@@ Line 96: / Line 96: @@
 We can define the height of a rational point simply as the Euclidean norm on its Plücker coordinates <math>X = \iota (P)</math>.
 :<math>
-	H(P) = \left\| X \right\| = \left\| p_1 \wedge \ldots \wedge p_n \right\| \\
+	H(P) = \left\| X \right\| = \left\| m_1 \wedge \ldots \wedge m_n \right\| \\
 </math>
-This is equivalent to the covolume of the lattice defined by P (also know as the lattice determinant), which can be easily computed using the Gram matrix.
+In terms of the lattice defined by P, this definition is equivalent to the volume of the {{w|fundamental domain}}, also known as the lattice determinant.
+It is easy to show that this does not depend on the basis we choose.
+The height can be easily computed using the {{w|Gram matrix}}:
 :<math>
 	\begin{align}
-	\mathrm{G}_{ij} &= \left\langle p_i, p_j \right\rangle \\
+	\mathrm{G}_{ij} &= \left\langle m_i, m_j \right\rangle \\
-	\sqrt{\det(\mathrm{G})} &= \left\| p_1 \wedge \ldots \wedge p_n \right\| = \left\| X \right\| \, .
+	\sqrt{\det(\mathrm{G})} &= \left\| m_1 \wedge \ldots \wedge m_n \right\| = \left\| X \right\| \, .
 	\end{align}
 </math>