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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
Citations:
Zbl 0691.14016
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References:
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