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Tangent cones to generalised theta divisors and generic injectivity of the theta map

Part of: Commutative algebra: Homological methods Cycles and subschemes Curves Local theory Special varieties

Published online by Cambridge University Press:  13 September 2017

George H. Hitching  and
Michael Hoff
George H. Hitching
Affiliation:
Høgskolen i Oslo og Akershus, Postboks 4, St. Olavs plass, 0130 Oslo, Norway email george.hitching@hioa.no
Michael Hoff
Affiliation:
Universität des Saarlandes, Campus E2 4, D-66123 Saarbrücken, Germany email hahn@math.uni-sb.de
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Abstract

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^{0}(C,E)=r+1$. For $g\geqslant (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the generalised theta divisor $\unicode[STIX]{x1D6E9}_{E}$ at ${\mathcal{O}}_{C}$. We use this to give a constructive proof and a sharpening of Brivio and Verra’s theorem that the theta map $\mathit{SU}_{C}(r){\dashrightarrow}|r\unicode[STIX]{x1D6E9}|$ is generically injective for large values of $g$.

Keywords

vector bundlesgeneralised theta divisorsBrill–Noether theory

MSC classification

Primary: 14H60: Vector bundles on curves and their moduli 14H51: Special divisors (gonality, Brill-Noether theory) 14M12: Determinantal varieties 14B10: Infinitesimal methods 13D10: Deformations and infinitesimal methods
Secondary: 14C34: Torelli problem
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 12 , December 2017 , pp. 2643 - 2657
Copyright
© The Authors 2017 

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