Algebraic Curves Lectures 26 And 27

Question: For what triples \((g,r,d)\) are there maps \(f:X\to \mathbb P^r\) such that \(f\) has \(\deg d\) and \(X\) has genus \(g\)?

Idea: Look at the locus of line bundles

\begin{align*} W^r_d(X) = \{L\mid r(L) \geq r\} \subseteq \operatorname{Pic}^d(X) \end{align*}

whose rank is at least \(r\).

We’ll need a few ingredients.

1. Determinantal varieties
2. Some facts about the Picard scheme \(\operatorname{Pic}^d(X)\)

Let \(M_K\) denote the locus of matrices of rank \(\leq k\), and note that this is a subset of \(M(n,m)\). This is called the generic determinantal variety. We can realize \(M_K\) as a variety in the following way. Define this auxiliary object

\begin{align*} \tilde{M}_k = \{(A, W) \in M(n,m) \times Gr(n-k, n) \mid A\cdot W = 0\} \end{align*}

consisting of all tuples \((A,W)\) where \(A\) is a \(n\times m\) matrix and \(W\) is a \((n-k)\)-dimensional subset of \(k^n\) where \(W\subset \ker A\). We then get a map \(\varphi:\tilde{M}_k \to M_k\), and it is generically \(1\) to \(1\): if \(A\in M_k\) has rank \(k\) then \(\varphi^{-1}(A) = (A, \ker A)\).

Let \(\pi_2\) denote the projection onto the second component:

\begin{align*} \pi_2:M(n,m)\times Gr(n-k,M) \to Gr (n-k,n). \end{align*}

For \(H\in Gr(n-k, n)\), the fiber of \(\pi_2|_{\tilde{M}_k}\) consists of all the \(A\in M(n,m)\) which vanish on \(H\). This is isomorphic to \(\mathbb A^{m - k}\) (to see this, row reduce \(A\)).

It is true that \(\tilde{M}_k\) is irreducible, and hence \(M_k\) is irreducible since \(\varphi\) is surjective. We (meaning those of us in class) don’t know why \(\tilde{M}_k\) is irreducible, it would follow from the properness of \(\pi_2\) except that \(\pi_2\) isn’t proper, so for now we simply accept that \(\tilde{M}_k\) is irreducible.

Proof

Fix \(H \in Gr(\ell - e, \ell)\) and let \(V\) be the \(\ell\)-dimensional vector space whose \(\ell-e\) dimensional subspaces \(Gr(\ell - e, \ell)\) parameterizes. Let \(\lambda\in (\bigwedge^e V)^*\) and \(U\subseteq Gr(e, \ell)\) be the set given by \(\lambda \neq 0\). Then \(U\) is the subset of \(e\)-dimensional subspaces transversal to \(H\). Any element in \(U\) can be identified with the graph of a linear function \(V/H \to H\). Elaborating: each choice of \(H\) gives you a split short exact sequence \(H\to V\to V/H\), which in turn gives you a map \(V/H \to H\). Conversely, if you have a surjective map \(V \to B\) with \(B\) isomorphic to \(V/H\), you get a subspace \(H\) in \(V\) so that \(H\oplus B \cong V\).

This means

\begin{align*} U \cong \Hom(V/H, H) \cong \mathbb C^{(\ell - e)\cdot e}. \end{align*}

Note that \((\ell - e)\cdot e\) is the product of the dimension of \(H\) with the dimension of an element in \(U\).

We have a Plucker embedding \(Gr(e, \ell) \hookrightarrow (\bigwedge^e V)^*\), and so these \(U\)’s actually give us the affine charts of \(Gr(e, \ell)\).

For a variety \(X\) over \(k\) and \(p\in X\) a point, we define the tangent space of \(X\) at \(p\) by

\begin{align*} T_pX = \Hom((\Spec k[\epsilon]/(\epsilon^2), (\epsilon)), (X, p)). \end{align*}

Example: Consider \(T_H Gr(k, V)\). Given a map

\begin{align*} \varphi:\Spec k[\epsilon]/(\epsilon^2) \to Gr (k,V) \end{align*}

and a basis \(\omega_1,...,\omega_k\) a basis of \(H\), we obtain a relative basis \(\omega_i + \epsilon\cdot v_i\). Conversely, a map \(\omega \mapsto v_i\) gives a linear map \(H\to V/H\). This should be enough to convince oneself that

\begin{align*} T_H Gr(k,V) = \Hom(H, V/H). \end{align*}

I arrived late and so these notes ought to be cross checked with Karl’s.

Last time we saw that \(T_WGr(\ell,V) \cong \Hom(W, V/W)\).

\begin{align*} T_{(W,W)}\widetilde{M}_k = \{(\varphi,B) ~\mid~ B\in M, \varphi\in \Hom(W,V/W) \text{ s.t. } B|_W = A\circ \varphi\}. \end{align*}

Take \(\pi:\widetilde{M}_k\to M_k\) to be the projection map. Then \(\pi_*T_{(A,W)}\widetilde{M}_k \subseteq T_AM_k\).

The forward direction is obvious, if \(B|_W = -A\circ \varphi\) then \(B\cdot W \subset \img(A)\). Likewise, if \(B\cdot W \subset \img(A)\), we can define a \(\varphi\) by setting for any \(B(w) = A(v)\) \(\varphi(w) = -v\). This defines a map \(\varphi:W\to V/W\).

\(\pi_* T_{(A,W)}\widetilde{M}_k = \{B\in M ~\mid~ B\cdot W\subseteq \img(A)\}.\)

\begin{align*} \dim(\pi_*T_{(A,W)}\widetilde{M}) &= \dim(\Hom(W,\img(A))) + k\cdot m \\ &= (n-k)k + k\cdot m = (n+m - k)k \\ &= \dim(M_k) = \dim(\widetilde{M}_k) \end{align*}