Verlinde bundles and generalized theta linear series (Q2781389)

Let \(X\) be a smooth projective complex curve of genus \(g\geq 2\), and denote by \(U_X(r,0)\) the moduli space of semi-stable vector bundles of rank \(r\) and degree 0 on \(X\). Let \(SU_X(r)\) be the moduli space of rank-\(r\) vector bundles on \(X\) with trivial determinant bundle. The purpose of the paper under review is to introduce and study the push-forward bundles of the pluri-theta line bundles on \(U_X(r,0)\) to the Jacobian \(J(X)\) by the determinant map \(\det: U_X(r,0)\to J(X)\). These bundles \(E_{r,k}\) on \(J(X)\) are called the Verlinde bundles (on the Jacobian), and they are precisely defined by \(E_{r,k} :=\det_* {\mathcal O}(k\cdot \Theta_N)\), where \(\Theta_N\) is the generalized theta divisor on \(U_X(r,0)\) associated with a line bundle \(N\in\text{Pic}^{g-1} (X)\). Then the fibers of those Verlinde bundles are precisely the well-knovm Verlinde spaces of level-\(k\) theta functions on \(SU_X(r)\), the dimension of which is determined by the famous Verlinde formula.NEWLINENEWLINENEWLINEThe author's strategy is to study the geometry of the Verlinde bundles \(E_{r,k}\) on the Jacobian \(J(X)\), which provides a more geometric approach to a complete understanding of the Verlinde spaces. In particular, the behavior of the Verlinde bundles under natural operations associated to vector bundles over abelian varieties is thoroughly analyzed, where special emphasis is put on their Fourier-Mukai transforms. This leads to some new duality theorems for certain bundles and their spaces of sections, on the one hand, and to explicit results in the study of effective global generation and normal generation for pluri-theta line bundles on \(U_X(r,0)\), on the other hand.NEWLINENEWLINENEWLINEAs to the duality results for Verlinde bundles obtained here, the author's theorems generalize some earlier formulae due to \textit{R. Donagi} and \textit{L. Tu} [Math. Res. Lett. 1, 345-357 (1994; Zbl 0847.14027)] and \textit{A. Beauville}, \textit{M. S. Narasimhan} and \textit{S. Ramanan} [J. Reine Angew. Math. 398, 169-179 (1989; Zbl 0666.14015)], the latter one being known as ``the strange duality at level one'', whereas the explicit bounds for global generatedness, ampleness, or very-ampleness of pluri-theta bundles on \(U_X(r,0)\) are seemingly new and highly interesting. Also, extending a construction method due to \textit{M. Raynaud} (1982), the author shows that the Fourier-Mukai transforms of the dual Verlinde bundles, restricted to certain embeddings of the curve \(X\) into its Jacobian \(J(X)\), provide new examples of base points in the linear system of the determinant bundle \(L\) on \(U_X(r,0)\). In the last section of the paper, it is explained how the foregoing methods and results could be extended to the moduli spaces \(U_X(r,d)\) of rank-\(r\) vector bundles of arbitrary degree \(d\) over a curve \(X\).NEWLINENEWLINENEWLINEAltogether, this is an important contribution towards the understanding of the geometry of moduli spaces of vector bundles over a curve, with a strong potential for generalization to vector bundles on smooth algebraic surfaces. The paper is very well written, lucid, detailed, rigorous and enlightening.