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Prym varieties and the Verlinde formula. (English) Zbl 0806.14026
In this paper, the Verlinde formula for vector bundles of rank 2 and even degree is considered. It is an explicit formula in terms of the genus and the degree to give the dimension of the space of sections of the bundle $$L^{\otimes k}$$ on $$\text{SM}_ C(2)$$, where $$C$$ is a curve (or Riemann surface) of genus $$g$$, $$\text{SM}_ C(2)$$ the moduli space of vector bundles of rank 2 and with trivial determinant, and $$L$$ the positive generator of $$\text{Pic(SM}_ C(2))$$ (a group which is known to be $$\simeq \mathbb{Z})$$. Call this dimension $$D_ k$$ and the dimension predicted by the formula $$N_ k$$.
The authors prove the inequality $$D_ 4 \geq N_ 4$$. (Note that meanwhile the formula $$D_ k = N_ k$$ is established in full generality by various authors and also for more general structure groups.) They relate the question to Prym varieties in the following way. By A. Beauville [Bull. Soc. Math. Fr. 116, No. 4, 431-448 (1988; Zbl 0691.14016)] sections of $$H^ 0 (\text{SM}_ C(2),L)$$ can be identified with theta functions of order 2, i.e. sections of $${\mathcal O}_{J(C)} (2 \Theta)$$, where $$(J(C), \Theta)$$ is the Jacobian of $$C$$. Call this vector space $$V$$. Thus we get a map $$\varphi : \text{SM}_ C(2) \to \mathbb{P}(V)$$, and $$\varphi$$ is compatible with the action of the group $$J(C) [2]$$ of points of order 2 on $$\text{SM}_ C(2)$$ and $$\mathbb{P}(V)$$. The action on $$\mathbb{P}(V)$$ is induced by the action of the Heisenberg group $$H$$ defined by the group $$J(C)[2]$$ and the Weil pairing $$E$$ on it (i.e. the central extension $$1 \to C^* \to H \to J(C) [2] \to 0$$ such that the commutator on $$H$$ induces the Weil pairing $$x \wedge y \mapsto (-1)^{E(x,y)}$$). Therefore, each $$x \in J(C)[2]$$ has 2 eigenspaces, $$\mathbb{P} (V_ x^ \pm)$$, both of dimension $$2^{g-1}-1$$.
Each such $$x$$ defines an unramified double covering $$C_ x @>\pi>> C$$ and the Prym variety $$P_ x = \ker (J(C_ x) @>N>>J (C))^ 0$$ (where $$N$$ is the norm map of $$\pi)$$. The choice of a line bundle $$z$$ on $$C$$ with $$z^{\otimes 2} \simeq x$$ gives a map $$\varphi_ x : \text{Ker} (N) \to \text{SM}_ C(2)$$; $$p \mapsto \pi_ * (p) \otimes z$$ (since $$\text{det} \pi_ * p = \text{det} \pi_ * {\mathcal O}_{C_ x} = x$$ for $$p \in \ker(N))$$.
The map $$\varphi \circ \psi_ x$$, restricted to $$P_ x$$, gives a map $$\varphi_ x : P_ x \to \mathbb{P} (V_ x)$$ into one of the 2 eigenspaces $$\mathbb{P} (V_ x^ \pm)$$. The authors show that $$\varphi_ x$$ is the natural map $P_ x \to K(P_ x) \subset \mathbb{P} H^ 0 (P_ x, {\mathcal O}_{P_ x} (2 \Theta_{P_ x})) \simeq \mathbb{P} (V_ x)$ $$(K(P_ x)$$ is the Kummer variety of $$P_ x)$$. Then they study the multiplication map $m_ k : \text{Sym}^ k (V) \to H^ 0 (\text{SM}_ C(2), L^{\otimes k})$ and observe that $\dim \ker (m_ k) \leq \dim \{F \in \text{Sym}^ k(V):\;F \text{ vanishes on all Prym } s\}.$ Using the Schottky-Jung relations and the Donagi relation the authors are able to give an upper bound of $$\dim \ker (m_ 4)$$ which gives the announced lower bound for $$D_ 4$$.
Reviewer: H.Kurke (Berlin)

##### MSC:
 14H60 Vector bundles on curves and their moduli 14H42 Theta functions and curves; Schottky problem 14K30 Picard schemes, higher Jacobians
Zbl 0691.14016
Full Text:
##### References:
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