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Theta‐characteristics on singular curves
Author(s) -
Piontkowski Jens
Publication year - 2007
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdm021
Subject(s) - mathematics , sheaf , line bundle , canonical bundle , pure mathematics , modulo , gravitational singularity , rank (graph theory) , bundle , genus , mathematical analysis , combinatorics , botany , biology , materials science , composite material
On a smooth curve a theta‐characteristic is a line bundle L , the square of which is the canonical line bundle ω. The equivalent condition ℋ om ( L , ω) ≅ L generalizes well to singular curves, as applications show. More precisely, a theta‐characteristic is a torsion‐free sheaf ℱ of rank 1 with ℋ om (ℱ, ω) ≅ ℱ. If the curve has non‐ADE singularities, then there are infinitely many theta‐characteristics. Therefore, theta‐characteristics are distinguished by their local type. The main purpose of this article is to compute the number of even and odd theta‐characteristics (that is ℱ with h 0 ( C , ℱ)≡ 0 and h 0 ( C , ℱ)≡ 1 modulo 2, respectively) in terms of the geometric genus of the curve and certain discrete invariants of a fixed local type.