cayleyTrick(I,k)
Let $X\subset\mathbb{P}^n$ be a $k$-dimensional projective variety. Consider the product $W = X\times\mathbb{P}^k$ as a subvariety of $\mathbb{P}(Mat(k+1,n+1))$, the projectivization of the space of $(k+1)\times (n+1)$-matrices, and consider the projection $p:\mathbb{P}(Mat(k+1,n+1))\dashrightarrow\mathbb{G}(k,n)=\mathbb{G}(n-k-1,n)$. Then the "Cayley trick" states that the dual variety $W^*$ of $W$ equals the closure of $p^{-1}(Z_0(X))$, where $Z_0(X)\subset\mathbb{G}(n-k-1,n)$ is the Chow hypersurface of $X$. The defining form of $W^*$ is also called the $X$-resultant. For details and proof, see Multiplicative properties of projectively dual varieties, by J. Weyman and A. Zelevinsky; see also Coisotropic hypersurfaces in Grassmannians, by K. Kohn.
In the example below, we apply the method to the quadric $\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^3$.
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In the next example, we calculate the defining ideal of $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1\subset\mathbb{P}^7$ and that of its dual variety.
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If the option Duality is set to true, then the method applies the so-called "dual Cayley trick".
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The object cayleyTrick is a method function with options.