Algebraic Curves Lecture 13

Fix a basis for \(V\). The coordinates of a \(k\)-dimensional subspace \(W\subseteq V\) are given by the \(K\times K\) minors of the \(n\times k\) matrix with columns basis vectors of \(W\).

For any two sequences \(i_1 < ... < i_{k-1}\) and \(j_1 < ... < j_{k+1}\) with \(i\leq i_s, j_\ell \leq n\), it holds that

\begin{align*} \sum^{k+1}_{\ell = 1}(-1)^{\ell} W_{i_1...i_{k-1}j_\ell} W_{j_1...\hat{j_\ell}...j_{k+1}} = 0 \end{align*}

where \(W_{i_1...i_{k-1}j_\ell}\) is the coordinate corresponding to the \(k\times k\) minor with rows in \(i_1...i_{k-1}j_\ell\) under the Plücker embedding. Such relations are called the Plücker relations.

Example: Consider the Plücker embedding \(\varphi: G(2,4) \to \bigwedge^2 V\) and choose \(W\in G(2,4)\). Choose a basis \(v_1,v_2\) of \(W\) where

\begin{align*} v_1 &= a_1e_1 + a_2e_2 + a_3e_3 + a_4e_4 \\ v_2 &= b_1e_1 + b_2e_2 + b_3e_3 + b_4e_4. \end{align*}

These are the image of the matrix

\begin{align*} M = \begin{pmatrix} a_1 & b_1 \\ a_2 & b_2 \\ a_3 & b_3 \\ a_4 & b_4 \end{pmatrix}. \end{align*}

Taking \(v_1\wedge v_2\) gives me

\begin{align*} v_1 \wedge v_2 &= (a_1e_1 + a_2e_2 + a_3e_3 + a_4e_4)\wedge (b_1e_1 + b_2e_2 + b_3e_3 + b_4e_4) \\ &= b_1a_2\cdot (e_2 \wedge e_1) + b_2\cdot a_1 \cdot (e_1 \wedge e_2) ... \\ &= (b_1 \cdot a_2 - b_2 \cdot a_1)(e_1\wedge e_2) + ... \end{align*}

Now take sequences \(i_1 = 1\) (only option) and \(j_1 = 1, j_2 = 2, j_3 = 3\) (\(k = 2\) and \(n=4\)). Then the Plücker relation is given

\begin{align*} - 0 + W_{12} \cdot W_{13} - W_{13}\cdot W_{12} = 0, \end{align*}

which is always true. Taking \(i_1 = 1\), \(j_1 = 2, j_3 = 3, j_4 = 4\) and we get

\begin{align*} - W_{12}\cdot W_{34} + W_{13}\cdot W_{24} - W_{14}\cdot W_{23} = 0, \end{align*}

which is a nonempty statement. This is the only Plücker relation in this case. This realizes \(G(2,4)\) as a quadric hypersurface in \(\mathbb P^5\).

Claim: \(\lambda \in \bigwedge^2 V\) is decomposable if and only if \(\lambda \wedge \lambda = 0\).

Proof

One direction is immediate. For the other direction, suppose \(\lambda\) is indecomposable and write it \(\lambda = e_1\wedge e_2 + e_3\wedge e_4\) for some choice of coordinates on \(V\). Then

\begin{align*} (\lambda \wedge \lambda) &= 2 (e_1\wedge e_2\wedge e_3 \wedge e_4) \neq 0. \end{align*}

Note that \(\lambda\wedge\lambda\) lives in \(\bigwedge^4 V\).

Let’s return to the above example of \(G(2,4)\) and the Plücker relation

\begin{align*} - W_{12}\cdot W_{34} + W_{13}\cdot W_{24} - W_{14}\cdot W_{23} = 0. \end{align*}

We now have a characterization of decomposable elements, so for

\begin{align*} \lambda &= \lambda_{12}(e_1\wedge e_2) + \lambda_{13}(e_1\wedge e_3) + \lambda_{14}(e_1\wedge e_4) \\ &\hspace{0.5cm} + \lambda_{23}(e_2\wedge e_3) + \lambda_{24}(e_2\wedge e_4) + \lambda_{34}(e_3\wedge e_4), \end{align*}

we get

\begin{align*} \lambda \wedge \lambda &= (\lambda_{12} \cdot \lambda_{34} - \lambda_{13}\cdot \lambda_{24}+\lambda_{14}\cdot \lambda_{23})(e_1\wedge e_2 \wedge e_3 \wedge e_4). \end{align*}

The coefficient \(\lambda_{12} \cdot \lambda_{34} - \lambda_{13}\cdot \lambda_{24}+\lambda_{14}\cdot \lambda_{23}\) gives the Plücker relation. Set

\begin{align*} \Sigma = \left\{(x,H) ~\middle|~ x\in H\right\} \subseteq \mathbb P^n\times G(k,\mathbb P^n) \end{align*}

and consider the projections \(\pi_1:\Sigma \to \mathbb P^n\) and \(\pi_2:\Sigma\to G(k,\mathbb P^n)\). We call \(\Sigma\) an incidence variety. It is a subvariety since the condition \(x \in H\) means that \(x \wedge b_1 \wedge ... \wedge b_k = 0\) where \(b_1,...,b_k\) is a basis for \(H\), and hence

\begin{align*} \Sigma = \left\{(x,h) ~\middle|~ x\wedge b_1 \wedge ... \wedge b_k = 0\right\} \end{align*}

meaning that \(\Sigma\) is a variety since \(x \wedge b_1 \wedge ... \wedge b_k\) is a polynomial equation.

For any subvariety \(Y\subseteq G(k,\mathbb P^n)\), I get a subvariety \(\pi_1(\pi_2^{-1}(Y))\) of \(\mathbb P^n\). If I take two subvarieties \(X,Y\subseteq \mathbb P^n\) and consider the rational map

\begin{align*} \varphi: X\times Y&\to G(1,\mathbb P^n) \\ (x,y)&\mapsto x\wedge y \end{align*}

defined outside the set \(x = y\) in \(X\times Y\), then we can define two new varieties

\begin{align*} J(X,Y) &= \pi_1(\pi_2^{-1}(\overline{\operatorname{img} \varphi})) \hspace{1cm}&\text{"join of \(X\) and \(Y\)"}\\ \textrm{Sec}(X) &= J(X,X)\hspace{1cm}&\text{"secant variety"}. \end{align*}

These have dimensions satisfying

\begin{align*} \dim(J(X,Y)) &\leq \dim (X) + \dim(Y) + 1\\ \dim(\textrm{Sec}(X)) &\leq 2\dim X + 1. \end{align*}

Example: Take \(X\) to be a curve in \(\mathbb P^n\) with \(n\geq 3\).

• \(\dim(\textrm{Sec}(X)) = 1\) if and only if \(X\) is a line.
• \(\dim(\textrm{Sec}(X)) = 2\) if and only if \(X\) is contained in a plane but not a plane.
• \(\dim(\textrm{Sec}(X)) = 3\) if and only if \(X\) is not contained in a plane.