Recent zbMATH articles in MSC 14-03https://www.zbmath.org/atom/cc/14-032021-04-16T16:22:00+00:00WerkzeugGenus 2 curves and generalized theta divisors.https://www.zbmath.org/1456.140392021-04-16T16:22:00+00:00"Brivio, Sonia"https://www.zbmath.org/authors/?q=ai:brivio.sonia"Favale, Filippo F."https://www.zbmath.org/authors/?q=ai:favale.filippo-francescoLet \({\mathcal U}_C(r,n)\) be the compactification of the moduli space of rank \(r\), degree \(n\) stable vector bundles on a projective complex curve \(C\), irreducible, smooth, of genus \(g \ge 2\). In the special case when \(n=r(g-1)\) it has a theta divisor \(\Theta _r\), defined as the ``natural Brill-Noether locus''.
For a fixed \(L \in \text{Pic}^{r(g-1)}(C)\) one has a moduli space of semistable vector bundles \({\mathcal S\mathcal U}_C(r,L)\) and a theta divisor \({\Theta }_{r,L}\).
Denote by \({\mathcal U}_C(r,n)\) the (compactification of) the moduli space (introduced by Seshadri) of rank \(r\), degree \(n\) stable vector bundles on \(C\). The main results of this paper are Theorems 2.5 and 3.4, for curves \(C\) of genus \(g\):
``There exists a vector bundle \(\mathcal V\) on \({\mathcal U}_C(r-1,r)\) of rank \(2r-1\) whose fibers at the point \([F]\in {\mathcal U}_C(r-1,r)\) is \(\mathrm{Ext}^1(F,{\mathcal O}_C)\). Let \({\mathbb P}({\mathcal V})\) be the associated projective bundle and \(\pi :{\mathbb P}({\mathcal V}) \rightarrow {\mathcal U}_C (r-1,r)\) the natural projection. Then the map \(\Phi : {\mathbb P}({\mathcal V}) \rightarrow {\Theta}_r\), sending \([v]\) to the vector bundle which is the extension of \(\pi ({v})\) by \({\mathcal O}_C\) , is a birational morphism''
and
``For a general stable bundle \(F \in {\mathcal S\mathcal U}_C(r,L)\) the map \[ \theta \circ \Phi \mid_{{\mathbb P}_F} : {\mathbb P}_F \rightarrow \mid r\Theta _M \mid \] is a linear embedding.''
In the second theorem \(\theta \) is the well known theta map and \({\mathbb P}_F=\pi ^{-1}([F])\).
The paper, which is very well written, explains the background, gives historical information and contains also other results which are of real interest.
Reviewer: Nicolae Manolache (Bucureşti)