COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$

Abstract

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (P r−1 ) rg //PGL(r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces. a double covering of P 8 branched along a hypersurface of degree six C 6 called the Coble-Dolgachev sextic [25]. Our result is the following. Theorem 1.3. The Coble-Dolgachev sextic C 6 is birational to a fibration over P 4 whose fibers are Igusa quartics. More precisely, C 6 contains a four-dimensional family of Igusa quartics parametrized by an open subset of P 4 .We recall that an Igusa quartic is a modular quartic hypersurface in P 4 that is related to some classical GIT quotients (see e.g. [15]) and moduli spaces. Its dual variety is a cubic three-fold called the Segre cubic, that is isomorphic to the GIT quotient (P 1 ) 6 //PGL(2).If r = 2 and g = 3, then SU C (2) is embedded by θ in P 7 as a remarkable quartic hypersurface C 4 called the Coble quartic [27]. Our methods also allow us to give a quick proof of the following fact.

Theta divisors and the geometry of tautological model

“…A general stable F ∈ U C (r, rg) satisfies the following conditions: h 0 (F) = r and F is generically globally generated, see [8]; this means that F fit into an exact sequence as follows:…”

“…is actually a rational surjective map, a general fiber is birational to the GIT quotient of (P r −1 ) rg with respect to the diagonal action of PG L(r ), see [8] for details. For any d ∈ C (g+1) let's consider the following subvariety of U C (r, rg): 1) ), so A d is irreducible and we have:…”

Nodal curves and polarizations with good properties

“…These spaces are interesting by themselves as higher dimensional varieties but also for important related constructions: just to mention some, one can consider higher-rank Brill-Noether theory, Theta divisors and Theta functions and the moduli spaces of coherent systems. For surveys on these topics see, for example [3,6,7]; for some results by the authors see [5,8,[11][12][13][14]. When the curve is singular, these spaces are not in general complete.…”

Coherent systems and BGN extensions on nodal reducible curves

“…For any real parameter α, the notion of α-stability has been introduced and it gives a family of coarse moduli spaces parametrizing coherent systems (E, V ) of type (r, d, k), i.e. with rk(E) = r, deg(E) = d and dim V = k. For comparison of different notions of stability in moduli theory see for instance [BB12].…”