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Instanton bundles on \(\mathbb{P}^1 \times \mathbb{F}_1\). (English) Zbl 1471.14029

The paper deals with a class of instanton bundles on the Fano threefold (of index one) \(F = \mathbb{P}^1 \times \mathbb{F}_1\) where \(\mathbb{F}_1 \subseteq \mathbb{P}^8\) is the degree \(8\) del-Pezzo surface obtained by blowing-up a point of \(\mathbb{P}^2\). In the first part of the paper, using the theory developed by A. L. Gorodentsev and S. A. Kuleshov [Helix Theory; Mosc. Math. J. 4, No. 2, 377–440 (2004; Zbl 1072.14020)], the authors showed that any instanton bundle on \(F\) (satisfying certain numerical criterion) can be realized as the cohomology in degree \(0\) of certain complexes. By an detained computations of cohomology, the authors have given a necessary and sufficient condition (Theorem 1.1) of the existence of instanton bundles on \(F\) under certain numerical criterion.
In the second part of the paper, the authors showed that under the aforementioned numerical criterion, a \(\mu\)-stable instanton bundle on \(F\) exists (Theorem 1.2). The proof is based on Hartshorne-Serre correspondence and a detail study of certain locally complete intersection curves on \(F\). It was further proved that the constructed instanton bundles are smooth points in the open subscheme of the moduli space of simple rank \(2\) bundles on \(F\) (with appropriate numerical conditions) consisting of instanton bundles.
Finally, the authors proved the existence of minimal instanton bundles on \(F\) and showed that in this cases the open subscheme of the moduli space of simple rank \(2\) bundles on \(F\) consisting of instanton bundles are smooth, irreducible and rational (Theorem 1.3). At the end, the authors discussed relations between minimal instanton bundles and weakly Ulrich bundles on a Fano three-fold of index \(\geq 3\).

MSC:

14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
14J60 Vector bundles on surfaces and higher-dimensional varieties, and their moduli
14J45 Fano varieties

Citations:

Zbl 1072.14020
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References:

[1] Altman, A. B.; Kleiman, S. L., Compactifying the Picard scheme, Adv. Math, 35, 1, 50-112 (1980) · Zbl 0427.14015
[2] Antonelli, V., Casnati, G., Genc, O.Even and odd instanton bundles on Fano threefolds. Work in progress. · Zbl 1485.14077
[3] Antonelli, V., Malaspina, F. (2020). Instanton bundles on the Segre threefold with Picard number three. Math. Nacht. 293:1026-1043. DOI: . · Zbl 1475.14016
[4] Arrondo, E., A home-made Hartshorne-Serre correspondence, Commun. Algebra, 20, 423-443 (2007) · Zbl 1133.14046
[5] Atiyah, M. F.; Hitchin, N. J.; Drinfeld, V. G.; Manin, Y. I., Construction of instantons, Phys. Lett. A, 65, 3, 185-187 (1978) · Zbl 0424.14004
[6] Atiyah, M. F.; Ward, R. S., Instantons and algebraic geometry, Commun. Math. Phys, 55, 2, 117-124 (1977) · Zbl 0362.14004
[7] Buchdahl, N. P., Stable 2-bundles on Hirzebruch surfaces, Math. Z, 194, 1, 143-152 (1987) · Zbl 0627.14028
[8] Buchdahl, N. P., Monads and bundles on rational surfaces, Rocky Mt. J. Math, 34, 2, 513-540 (2004) · Zbl 1061.14039
[9] Casnati, G., Coskun, E., Genc, O., Malaspina, F. (2021). Instanton bundles on the blow up of \(####\) at a point. Mich. Math. J. DOI: . · Zbl 1490.14067
[10] Casnati, G.; Genc, O., Instanton bundles on two Fano threefolds of index 1, Forum Math, 32, 5, 1315-1336 (2020) · Zbl 1485.14077
[11] Eisenbud, D.; Schreyer, F. O.; Weyman, J., Resultants and Chow forms via exterior syzigies, J. Am. Math. Soc, 16, 3, 537-579 (2003) · Zbl 1069.14019
[12] Faenzi, D., Even and odd instanton bundles on Fano threefolds of Picard number one, Manuscripta Math, 144, 1-2, 199-239 (2014) · Zbl 1293.14009
[13] Gorodentsev, A. L.; Kuleshov, S. A., Helix theory, MMJ, 4, 2, 377-440 (2004) · Zbl 1072.14020
[14] Hartshorne, R., Algebraic Geometry (1977), Berlin, Heidelberg: Springer, Berlin, Heidelberg
[15] Hartshorne, R., Coherent functors, Adv. Math, 140, 1, 44-94 (1998) · Zbl 0921.13010
[16] Huybrechts, D.; Lehn, M., The Geometry of Moduli Spaces of Sheaves (2010), Cambridge, MA: Cambridge University Press, Cambridge, MA · Zbl 1206.14027
[17] Iskovskikh, V. A.; Yu, G.; Parshin, A. N.; Shafarevich, I. R., Algebraic Geometry V, Prokhorov: Fano varieties (1999), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 0903.00014
[18] Jardim, M. B.; Menet, G.; Prata, D. M.; Sá Earp, H. N., Holomorphic bundles for higher dimensional gauge theory, Bull. London Math. Soc, 49, 1, 117-132 (2017) · Zbl 1386.14063
[19] King, A., Instanton and holomorphic bundles on the blown-up plane (1989)
[20] Kuznetsov, A., Instanton bundles on Fano threefolds, Centr. Eur. J. Math, 10, 4, 1198-1231 (2012) · Zbl 1282.14075
[21] Malaspina, F., Marchesi, S., Pons-Llopis, J. (2020). Instanton bundles on the flag variety \(####\) Ann. Sc. Nor. Super. Pisa Cl. Sci. XX(4):1469-1505. DOI: . · Zbl 1475.14029
[22] Maruyama, M., Boundedness of semistable sheaves of small ranks, Nagoya Math. J, 78, 65-94 (1980) · Zbl 0456.14011
[23] Mumford, D., Lectures on Curves on an Algebraic Surface. With a Section by G. M. Bergman (1966), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 0187.42701
[24] Okonek, C.; Schneider, M.; Spindler, H., Vector Bundles on Complex Projective Spaces (1980), Berlin, Heidelberg: Springer, Berlin, Heidelberg · Zbl 0438.32016
[25] Orlov, D. O., Projective bundles, monoidal transformations, and derived categories of coherent sheaves, Russian Acad. Sci. Izv. Math. Russian Acad. Sci. Izv. Math, 41, 1, 133-141 (1993) · Zbl 0798.14007
[26] Ottaviani, G., Some extensions of Horrocks criterion to vector bundles on Grassmannians and quadrics, Ann. Mat. Pura Appl, 155, 1, 317-341 (1989) · Zbl 0718.14010
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