Very ample vector bundles of curve genus two

Let C be a curve of genus two. We denote by [Formula: see text] the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over C, and by Jd the variety of line bundles of degree d on C. In particular, J1 has a canonical theta divisor Θ. The space [Formula: see text] is a double cover of ℙ8 = |3Θ| branched along a sextic hypersurface, the Coble sextic. In the dual [Formula: see text], where J1 is embedded, there is a unique cubic hypersurface singular along J1, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre–Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.

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Ample Vector Bundles of Curve Genus One

Canadian Mathematical Bulletin ◽

10.4153/cmb-1999-025-9 ◽

1999 ◽

Vol 42 (2) ◽

pp. 209-213 ◽

Cited By ~ 3

Author(s):

Antonio Lanteri ◽

Hidetoshi Maeda

Keyword(s):

Vector Bundle ◽

Vector Bundles ◽

Projective Variety ◽

Ample Vector Bundle ◽

Smooth Complex ◽

Complex Projective Variety ◽

Curve Genus ◽

Genus One ◽

Complex Projective ◽

Zero Locus

AbstractWe investigate the pairs (X, ε) consisting of a smooth complex projective variety X of dimension n and an ample vector bundle ε of rank n − 1 on X such that ε has a section whose zero locus is a smooth elliptic curve.

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RANK FOUR SYMPLECTIC BUNDLES WITHOUT THETA DIVISORS OVER A CURVE OF GENUS TWO

International Journal of Mathematics ◽

10.1142/s0129167x08004716 ◽

2008 ◽

Vol 19 (04) ◽

pp. 387-420 ◽

Cited By ~ 2

Author(s):

GEORGE H. HITCHING

Keyword(s):

Line Bundle ◽

Moduli Space ◽

Vector Bundles ◽

Projective Variety ◽

Theta Divisor ◽

Weierstrass Points ◽

Base Locus ◽

Canonical Bijection ◽

Genus Two ◽

Irreducible Projective Variety

The moduli space [Formula: see text] of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier–Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from [Formula: see text] to the space of even 4Θ divisors on the Jacobian variety of the curve.

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