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Genus 2 curves and generalized theta divisors. (English) Zbl 1456.14039

Let \({\mathcal U}_C(r,n)\) 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.

MSC:

14H60 Vector bundles on curves and their moduli
14H51 Special divisors on curves (gonality, Brill-Noether theory)
14H40 Jacobians, Prym varieties
14-03 History of algebraic geometry
01A60 History of mathematics in the 20th century
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References:

[1] Atiyah, M. F., Complex analytic connections in fibre bundles, Trans. Am. Math. Soc., 85, 181-207 (1957) · Zbl 0078.16002
[2] Arbarello, E.; Cornalba, M.; Griffiths, P. A.; Harris, J., Geometry of Algebraic Curves, I (1985), Springer Verlag: Springer Verlag Berlin · Zbl 0559.14017
[3] Beauville, A., Some stable vector bundles with reducible theta divisors, Manuscr. Math., 110, 343-349 (2003) · Zbl 1016.14016
[4] Beauville, A., Vector bundles and the theta functions on curves of genus 2 and 3, Am. J. Math., 128, n3, 607-618 (2006) · Zbl 1099.14024
[5] Basu, S.; Dan, K., Stability of secant bundles on the second symmetric power of curves, Arch. Math. (Basel), 110, 245-249 (2018) · Zbl 1387.14046
[6] Beauville, A.; Narasimhan, M. S.; Ramanan, S., Spectral curves and the generalised theta divisor, J. Reine Angew. Math., 398, 169-178 (1989) · Zbl 0666.14015
[7] Biswas, I.; Nagaraj, D. S., Reconstructing vector bundles on curves from their direct image on symmetric powers, Arch. Math. (Basel), 99, 4, 327-331 (2012) · Zbl 1257.14030
[8] Brivio, S., Families of vector bundles and linear systems of theta divisors, Int. J. Math., 28, 6, Article 1750039 pp. (2017), (16 pages) · Zbl 1371.14036
[9] Brivio, S.; Verra, A., The Brill Noether curve of a stable vector bundle on a genus two curve, (Nagel, J.; Peters, C., Algebraic Cycles and Motives. Algebraic Cycles and Motives, London Math. Soc. LNS, vol. 344, v 2 (2007), Cambridge Univ. Press) · Zbl 1184.14058
[10] Drezet, I. M.; Narasimhan, M. S., Groupe de Picard des variétés de modules de fibrés semi-stable sur les courbes algébriques, Invent. Math., 97, 53-94 (1989) · Zbl 0689.14012
[11] Ghione, F., Quelques résultats de Corrado Segre sur les surfaces réglées, Math. Ann., 255, 77-96 (1981) · Zbl 0435.14010
[12] King, A.; Schofield, A., Rationality of moduli of vector bundles on curves, Indag. Math. (N.S.), 10, 4, 519-535 (1999) · Zbl 1043.14502
[13] Laszlo, Y., Un théoréme de Riemann puor les diviseurs thetá sur les espaces de modules de fibrés stables sur une courbe, Duke Math. J., 64, 333-347 (1991) · Zbl 0753.14023
[14] Le Potier, J., Lectures on Vector Bundles (1997), Cambridge Univ. Press · Zbl 0872.14003
[15] Lange, H., Hohere Sekantenvarietaten und Vektordundel auf Kerven, Manuscr. Math., 52, 63-80 (1985) · Zbl 0588.14030
[16] Lange, H.; Narasimhan, M. S., Maximal subbundles of rank two vector bundles on curves, Math. Ann., 266, 55-72 (1983) · Zbl 0507.14005
[17] Lange, H.; Newstead, P. E., Maximal subbundles and Gromov-Witten invariants. A tribute to C.S. Seshadri, (Chennai, 2002. Chennai, 2002, Trends Math. (2003), Birkhäuser: Birkhäuser Basel), 310-322 · Zbl 1071.14036
[18] Maruyama, M., Elementary transformations in the theory of algebraic vector bundles, Lect. Notes Math., 961, 241-266 (1982) · Zbl 0505.14009
[19] Narasimhan, M. S.; Ramanan, S., Moduli of vector bundles on a compact Riemann surface, Ann. Math., 89, 2, 14-51 (1969) · Zbl 0186.54902
[20] Newstead, P. E., Rationality of moduli spaces of stable bundles, Math. Ann., 215, 251-268 (1975) · Zbl 0288.14003
[21] Okonek, C.; Teleman, A., Gauge theoretical equivariant Gromov-Witten invariants and the full Seiberg-Witten invariants of ruled surfaces, Commun. Math. Phys., 227, 3, 551-585 (2002) · Zbl 1037.57025
[22] Ortega, A., On the moduli space of rank 3 vector bundles on a genus 2 curve and the Coble cubic, J. Algebraic Geom., 14, 327-356 (2005) · Zbl 1075.14031
[23] Oxbury, W. M., Varieties of maximal line subbundles, Math. Proc. Camb. Philos. Soc., 129, 9-18 (2000) · Zbl 0973.14015
[24] Ramanan, S., The moduli spaces of vector bundles over an algebraic curve, Math. Ann., 200, 69-84 (1973) · Zbl 0239.14013
[25] Russo, B.; Teixidor i. Bigas, M., On a conjecture of lange, J. Algebraic Geom., 8, 483-496 (1999) · Zbl 0942.14013
[26] Schwarzenberger, R. L.E., The secant bundle of a projective variety, Proc. Lond. Math. Soc. (3), 14, 369-384 (1964) · Zbl 0123.38201
[27] Seshadri, C. S., Fibrés vectorials sur les courbes algébriques, Astérisque, 96 (1992)
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