Analytic moduli spaces of simple sheaves on families of integral curves

Abstract

We prove the existence of fine moduli spaces of simple coherent sheaves on families of irreducible curves. Our proof is based on the existence of a universal upper bound of the Castelnuovo-Mumford regularity of such sheaves, which we provide.

Vector bundles on degenerations of elliptic curves and Yang–Baxter equations

“…The first part of this theorem can be found in [2] the second statement is trivial. The third part seems to be well-known, see for example [52] for the case of an elliptic curve and [21] for the proof of a more general statement and further details.…”

“…An associative r-matrix is the germ of a meromorphic function in four variables r : C 4 (v 1 ,v 2 ;y 1 ,y 2 ) , 0 −→ Mat n×n (C) ⊗ Mat n×n (C) holomorphic on C 4 \ V (y 1 − y 2 )(v 1 − v 2 ) , 0 and satisfying the equation ( 4) r(v 1 , v 2 ; y 1 , y 2 ) 12 r(v 1 , v 3 ; y 2 , y 3 ) 23 = r(v 1 , v 3 ; y 1 , y 3 ) 13 r(v 3 , v 2 ; y 1 , y 2 ) 12 + +r(v 2 , v 3 ; y 2 , y 3 ) 23 r(v 1 , v 2 ; y 1 , y 3 ) 13 . Such a matrix is called unitary if (5) r(v 1 , v 2 ; y 1 , y 2 ) 12 = −r(v 2 , v 1 ; y 2 , y 1 ) 21 .…”

“…Tensoring the exact sequence (21) with the vector bundle W and applying Hom E (V, − ) we obtain an exact sequence…”