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Section 1.4 Rational Maps into Abelian Varieties (Maria)

Note all varieties are irreducible today.

Subsection 1.4.1 Rational maps

\(V,W\) varieties \(/K\text{.}\) Consider pairs \((U,\phi_U)\text{,}\) where \(\emptyset \ne U \subset V\) an open subset so \(U\) is dense, and \(\phi_U \colon U \to W\) is a regular map.

Definition 1.4.1. Rational maps.

\((U,\phi_U)\text{,}\) \((U',{\phi_{U'}})\) are equivalent if \(\phi_U\) and \(\phi_{U'}\) agree on \(U \cap U'\text{.}\) An equivalence class \(\phi\) of \(\{(U, \phi_U)\}\) is a rational map \(\phi \colon V \dashrightarrow W\) If \(\phi\colon V \dashrightarrow W\) is defined at \(v\in V\) if \(v\in U\) for some \((U,\phi_U) \in \phi\text{.}\)

Note 1.4.2.

The set \(U_1 = \bigcup U\) where \(\phi\) is defined is open and \((U_1,\phi_1) \in \phi\) where \(\phi_1 \colon U_1 \to W\) restricts to \(\phi_U\) on \(U\text{.}\)

Example 1.4.3.
  1. Let \(\emptyset \ne W \subseteq V\) be open. Then the rational map \(V\dashrightarrow W\) induced by \(\id \colon W \to W\) will not extend to \(V\text{.}\) To avoid this, assume \(W \) is complete (so \(W = V\)).
  2. \(C \colon y^2 = x^3\text{,}\) then \(\alpha\colon \aff^1 \to C\text{,}\) \(a\mapsto (a^2,a^3)\) is a regular map, restricting to an isomorphism \(\aff^1 \smallsetminus \{0 \} \to C \smallsetminus \{0\}\text{.}\) The inverse of \(\alpha|_{\aff^1\smallsetminus \{0\}}\) represents \(\beta \colon C \dashrightarrow \aff^1\) which does not extend to \(C\text{.}\) This corresponds on function fields to
    \begin{equation*} K(t) \to K(x,y) \end{equation*}
    \begin{equation*} t\mapsto y/x \end{equation*}
    which does not send \(K[y]_{(t)}\) to \(K[x,y]_{(x,y)}\text{.}\)
  3. Given a nonsingular surface \(V,\, P\in V\) then \(\exists \alpha\colon W \to V\) regular that induces an isomorphism \(\alpha\colon W\smallsetminus \alpha^{-1} (P) \to V\smallsetminus P\text{,}\) but \(\alpha^{-1}(P)\) is a projective line. The rational map represented by \(\alpha^{-1}\) is not regular on \(V\) (where to send \(P\text{?}\)).

(\(V\) a curve) \(V\) nonsingular curve, \(\emptyset\ne U\subseteq V\) open, \(\phi\colon U \to W\) a regular map.

\begin{equation*} \xymatrix{ & V\\ U\ar[ur]\ar[dr] \ar[r] & U' \subseteq Z \subseteq V\times W \ni (v,w)\ar[d]_q \ar[u]_p \\ & W \ni w } \end{equation*}

\(U'\) is the image of \(U\text{,}\) \(Z = \overline{U'}\text{.}\) \(W \) is complete, \(Z\) closed implies \(p(Z) \subseteq V\) is closed. Also, \(U \subseteq p(Z) \implies p(Z)= V\text{.}\)

\begin{equation*} U\xrightarrow{\sim} U' \to U \end{equation*}

so

\begin{equation*} U' \xrightarrow{\sim} U \end{equation*}
\begin{equation*} Z \twoheadrightarrow V \end{equation*}

this implies \(Z \xrightarrow\sim V\text{.}\) Then \(q|_Z \colon Z \to W\) is the extension of \(\phi \) to \(V\text{.}\)

Fix \((U, \phi_U) \in \phi\) and consider

\begin{equation*} \Phi\colon V\times V \dashrightarrow G \end{equation*}

represented by

\begin{equation*} U\times U\xrightarrow{\phi_U\times\phi_U} G\times G \xrightarrow{\id\times\operatorname{inv}} G\times G \xrightarrow{m} G \end{equation*}
\begin{equation*} (x,y) \mapsto \phi_U(x) \phi_U(y)^{-1} \end{equation*}

Check \(\phi\) is defined at \(x\) iff \(\Phi\) is defined at \((x,x)\) (and in this case \(\Phi(x,x) = e\)). This is equivalent to the map \(\Phi^*\colon \sheaf O_{G,e} \to K(V\times V)\) induced by \(\Phi\) satisfying \(\im(\sheaf O_{G,e}) \subseteq \sheaf O_{V\times V, (x,x)}\) For a nonzero function \(f\) on \(V\times V\text{,}\) write \(\divisor(f) = \divisor(f)_0 - \divisor(f)_\infty\) which are effective divisors. Then

\begin{equation*} \sheaf O_{V\times V, (x,x)} = \{0\} \cup \{f\in K(V\times V) : \divisor(f)_\infty \text{ does not contain }(x,x)\}\text{.} \end{equation*}

Suppose \(\phi\) is not defined at \(x\text{,}\) then there exists \(f\in \im(\sheaf O_{G,e})\) s.t. \((x,x) \in \divisor(f)_\infty\text{.}\) Then \(\Phi\) is not defined at any \((y,y) \in \Delta \cap \divisor(f)_\infty = \divisor(f^{-1})_0\text{,}\) which is a pure codimension 1 subset of \(\Delta\) by Milne's AG thm 9.2. The corresponding subset in \(V\) is of pure codimension 1, and \(\phi\) is not defined there.

Since group varieties are nonsingular, \(\alpha\colon G \to A\) is a regular map by Theorem 1.4.5. The rest is as proof of Corollary 1.2.

Subsection 1.4.2 Dominating and birational maps

Definition 1.4.9. Dominating maps.

\(\phi\colon V \dashrightarrow W\) is dominating if \(\im (\phi_U)\) is dense in \(W\) for a representative \((U,\phi_U)\in \phi\text{.}\)

Exercise: A dominating \(\phi\colon V\dashrightarrow W\) defines a homomorphism \(K(W) \to K(V)\) and any such homomorphism arises from a unique dominating rational map.

Definition 1.4.10.

\(\phi\colon V \dashrightarrow W\) is birational if the corresponding \(K(W)\to K(V)\) is an isomorphism or, equivalently if there exists \(\psi\colon W\dashrightarrow V \) s.t. \(\phi\circ \psi\) and \(\psi\circ\phi\) are the identity wherever they are defined. In this case we say \(V\) and \(W\) are birationally equivalent.

Note 1.4.11.

In general birational equivalence does not imply isomorphic. E.g. \(V\) a variety \(\emptyset \ne W\subsetneq V\) an open subset, or \(V= \aff^1, W \colon y^2 =x^3\text{.}\)

\(A,B\) abelian varieties with \(\phi \colon A\dashrightarrow B\) a birational map with inverse \(\psi\text{.}\) Then by Theorem 1.4.5 \(\phi ,\psi\) extend to regular maps \(\phi\colon A \to B \text{,}\) \(\psi\colon B\to A\) and \(\phi\circ\psi,\psi\circ\phi\) are the identity everywhere. This implies that \(\phi\) is an isomorphism of algebraic varieties and after composition with a translation, \(\phi\) is also a group isomorphism.

Theorem 1.4.5 implies \(\alpha \colon \aff^1 \dashrightarrow A\) extends to \(\alpha \colon \aff^1\to A\) and we may assume \(\alpha(0) = e\text{.}\) \((\aff^1, +)\text{:}\) \(\alpha(x+y) = \alpha(x) + \alpha(y)\) for all \(x,y\in \aff^1(K) = K\text{.}\) \((\aff^1\smallsetminus\{0\}, \cdot)\text{:}\) \(\alpha(xy) = \alpha(x) + \alpha(y) + c\) for all \(x,y\in K^\times\text{.}\) These can only hold at the same time if \(\alpha\) is constant. \(\PP^1 \dashrightarrow A\) is constant, since its constant on affine patches.

Definition 1.4.14.

\(V/\overline K\) is unirational if there is a dominating map \(\aff^n \dashrightarrow V\text{,}\) where \(n = \dim_{\overline K} V\text{.}\) \(V/K\) is unirational if \(V/K\) is.

Wlog \(K = \overline K\text{.}\) Since \(V\) is unirational we get \(\beta \colon \PP^1\times\cdots\times\PP^1 \dashrightarrow V\dashrightarrow A\text{,}\) which extends to \(\beta \colon \PP^1\times\cdots\times\PP^1 \to A\text{.}\) Then by Milne corollary 1.5, there exist regular maps \(\beta_i \colon \PP^1 \to A\) s.t. \(\beta(x_1,\ldots, x_n) = \sum \beta_i(x_i)\) and by Proposition 1.4.13 each \(\beta_i\) map is constant.