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Pencils of quadrics and hyperelliptic curves in characteristic two. (English) Zbl 0693.14015

We study hyperelliptic curves in characteristic 2. We show that there is a bijective correspondence between {connected hyperelliptic curves of genus g} and {orbits of a certain group acting on \[ \Gamma({\mathcal O}_{{\mathbb{P}}^ 1}(g+1))\times \Gamma ({\mathcal O}_{{\mathbb{P}}^ 1}(2g- 2))\}. \] We determine a standard form for regular pencils of quadrics (of characteristic 2) whose Pfaffian has all its roots distinct. To such a pencil we associate a hyperelliptic curve X. Unlike in characteristic zero, this curve is either nonsingular or nodal, it is nonsingular iff the intersection of quadrics of the pencil is nonsingular. In case the curve is nonsingular, we establish a bijective correspondence between bundles with \({\mathbb{Z}}/2\)-action on X and parabolic bundles on \({\mathbb{P}}^ 1\). Using this correspondence, we prove the following theorems (with their generalisations) in characteristic 2.
Theorem 1: The Jacobian of a nonsingular hyperelliptic curve X of genus g with \((g+1)\) Weierstrass points is isomorphic to the variety of g- dimensional subspaces of \(k^{2g+2}\) contained in the intersection of the quadrics \[ q_ 1\equiv \sum^{g+1}_{i=1}X_ iY_ i =0,\quad q_ 2\equiv \sum^{g+1}_{i=1}(a_ iX_ iY_ i+c_ iX_ i^ 2+d_ iY_ i^ 2) =0. \] Theorem 2: The moduli space of stable vector bundles of rank 2 and fixed determinant of odd degree on X (as above) is isomorphic to the variety of \((g-1)\)-dimensional subspaces of \(k^{2g+2}\) contained in the intersection of \(q_ 1\) and \(q_ 2.\)
In characteristic \(\neq 2\), these theorems were proved by Newstead \((g=2)\), M. Reid theorem 1), Desale and Ramanan (theorems 1 and 2).
Reviewer: U.Bhosle

MSC:

14H10 Families, moduli of curves (algebraic)
14G15 Finite ground fields in algebraic geometry
14H25 Arithmetic ground fields for curves
14H55 Riemann surfaces; Weierstrass points; gap sequences
14H40 Jacobians, Prym varieties
14D20 Algebraic moduli problems, moduli of vector bundles
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