Nội dung được dịch bởi AI, chỉ mang tính chất tham khảo
Các bó vector nửa ổn định trên các dạng biến hình
Tóm tắt
Chúng tôi giới thiệu và nghiên cứu khái niệm nửa ổn định liên quan đến các phép biến hình $$Y\xrightarrow {\scriptscriptstyle \pi } X$$ giữa các đa tạp dự án, khác với khái niệm của Maruyama–Simpson. Đối với các bó sheaf không xoắn trên Y và cho các phân cực thích hợp, chúng tôi liên kết tính (nửa) ổn định trên Y với độ dốc (nửa) ổn định trên sợi đại diện chung của $$\pi$$. Chúng tôi áp dụng kết quả này để miêu tả một cách chi tiết các không gian moduli của các bó vector nửa ổn định trên các mặt Hirzebruch và trên các ba chiều là các bó $$\mathbb P^2$$ -bundles trên. Chúng tôi thiết lập tính không thể phân tích và tính hữu tỉ của chúng, một vấn đề đã được nghiên cứu sâu sắc.
Từ khóa
#nửa ổn định #bó vector #biến hình #không gian moduli #đa tạp dự ánTài liệu tham khảo
Almeida, C., Jardim, M., Tikhomirov, A., Tikhomirov, S.: New moduli components of rank 2 bundles on projective space (2017). arXiv:1702.06520
Aprodu, M., Brînzănescu, V., Marchitan, M.: Rank-two vector bundles on Hirzebruch surfaces. Cent. Eur. J. Math. 10(4), 1321–1330 (2012)
Arrondo, E.: A home-made Hartshorne-Serre correspondence. Rev. Mat. Complut. 20(2), 423–443 (2007)
Artamkin, I.V.: Stable bundles with \(c_1 =0\) on rational surfaces. Math. USSR-Izv. 36(2), 231–246 (1991)
Barth, W.: Moduli of vector bundles on the projective plane. Invent. Math. 42, 63–91 (1977)
Barth, W., Hulek, K.: Monads and moduli of vector bundles. Manuscripta Math. 25(4), 323–347 (1978)
Bartocci, C., Bruzzo, U., Rava, C.L.S.: Monads for framed sheaves on Hirzebruch surfaces. Adv. Geom. 15(1), 55–76 (2015)
Brosius, J.E.: Rank-\(2\) vector bundles on a ruled surface I. Math. Ann. 265(2), 155–168 (1983)
Brosius, J.E.: Rank-\(2\) vector bundles on a ruled surface, II. Math. Ann. 266(2), 199–214 (1983)
Buchdahl, N.P.: Stable \(2\)-bundles on Hirzebruch surfaces. Math. Z. 194(1), 143–152 (1987)
Costa, L., Miró-Roig, R.M.: Rationality of moduli spaces of vector bundles on Hirzebruch surfaces. J. Reine Angew. Math. 509, 151–166 (1999)
Costa, L., Miró-Roig, R.M.: Moduli spaces of vector bundles on higher dimensional varieties. Michigan Math. J. 49(3), 605–620 (2001)
Costa, L., Miró-Roig, R.M.: Elementary transformations and the rationality of the moduli spaces of vector bundles on \({\mathbb{P}}^2\). Manuscripta Math. 113(1), 69–84 (2004)
Coandă, I., Tikhomirov, A., Trautmann, G.: Irreducibility and smoothness of the moduli space of mathematical \(5\)-instantons over \({\mathbb{P}}^3\). Internat. J. Math. 14(1), 1–45 (2003)
Domokos, M.: Covariants and the no-name lemma. J. Lie Theory 18(4), 839–849 (2008)
Donagi, R.Y.: Principal bundles on elliptic fibrations. Asian J. Math. 1(2), 214–223 (1997)
Donaldson, S.K.: Instantons and geometric invariant theory. Comm. Math. Phys. 93(4), 453–460 (1984)
Ellingsrud, G.: Sur l’irréductibilité du module des fibrés stables sur \(\mathbf{P^2}\). Math. Z. 182(2), 189–192 (1983)
Ellingsrud, G., Strømme, S.A.: Stable rank-2 vector bundles on \(\mathbf{P^3}\) with \(c_1=0\) and \(c_2=3\). Math. Ann. 255(1), 123–135 (1981)
Ellingsrud, G., Strømme, S.A.: On the rationality of the moduli space for stable rank-2 vector bundles on \({\mathbb{P}}^ 2\). In: Greuel, G., et al. (eds.) Singularities, Representation of Algebras, and Vector Bundles. Lecture Notes in Mathematics, vol. 1273, pp. 363–371. Springer, Berlin (1987)
Friedman, R.: Rank two vector bundles over regular elliptic surfaces. Invent. Math. 96(2), 283–332 (1989)
Friedman, R., Morgan, J., Witten, E.: Vector bundles over elliptic fibrations. J. Algebraic Geom. 8(2), 279–401 (1999)
Gieseker, D., Li, J.: Moduli of high rank vector bundles over surfaces. J. Amer. Math. Soc. 9(1), 107–151 (1996)
Greb, D., Toma, M.: Compact moduli spaces for slope-semistable sheaves. Algebr. Geom. 4(1), 40–78 (2017)
Grothendieck, A.: Revêtements Étales et Groupe Fondamental (SGA 1). In: Séminaire de Gémétrie Algébrique du Bois Marie 1960–61. Enlarged by two reports of M. Raynaud. Documents Mathmatiques, vol. 3. Société Mathématique de France, Paris (2003)
Hartshorne, R.: Algebraic Geometry, 8th edn. Springer, New York (1997)
Hartshorne, R.: Stable vector bundles of rank \(2\) on \({\mathbb{P}}^3\). Math. Ann. 238(3), 229–280 (1978)
Hoppe, H., Spindler, J.: Modulräume stabiler \(2\)-Bündel auf Regelflächen. Math. Ann. 249, 127–140 (1980)
Hulek, K.: On the classification of stable rank-\(r\) vector bundles over the projective plane. In: Hirschowitz, A. (ed.) Vector Bundles and Differential Equations. Progress in Mathematics, vol. 7, pp. 113–144. Birkhäuser, Boston (1980)
Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves, 2nd edn. Cambridge Mathematical Library. Cambridge University Press, New York (2010)
Katsylo, P.I.: Birational geometry of moduli varieties of vector bundles over \({\mathbb{P}}^2\). Math. USSR-Izv. 38(2), 419–428 (1992)
Kuznetsov, A.: Instanton bundles on Fano threefolds. Cent. Eur. J. Math. 10(4), 1198–1231 (2012)
Langer, A.: Semistable sheaves in positive characteristic. Ann. Math. 159(1), 251–276 (2004) (Addendum in Ann. Math. 160(3), 1211–1213 (2004))
Li, W.-P., Qin, Z.: Stable vector bundles on algebraic surfaces. Trans. Amer. Math. Soc. 345(2), 833–852 (1994)
Li, W.-P.: Relations between moduli spaces of stable bundles over \({\mathbb{P}}^2\) and rationality. J. Reine Angew. Math. 484, 210–217 (1997)
Maeda, T.: An elementary proof of the rationality of the moduli space for rank \(2\) vector bundles on \({\mathbb{P}}^2\). Hiroshima Math. J. 20(1), 103–107 (1990)
Markushevich, D., Tikhomirov, A.S.: Rationality of instanton moduli (2010). arXiv:1012.4132
Maruyama, M.: Moduli of stable sheaves I. J. Math. Kyoto Univ. 17(1), 91–126 (1977)
Maruyama, M.: Moduli of stable sheaves II. J. Math. Kyoto Univ. 18(3), 557–614 (1978)
Maruyama, M.: The rationality of the moduli spaces of vector bundles of rank 2 on \({\mathbb{P}}^2\). In: Algebraic Geometry. Advanced Studies in Pure Mathematics, vol. 10, pp. 399–414. North-Holland, Amsterdam (1987)
Nakashima, T.: On the moduli of stable vector bundles on a Hirzebruch surface. Math. Z. 212(2), 211–221 (1993)
Nakashima, T.: Moduli of vector bundles on Fano fibrations. JP J. Geom. Topol. 3(1), 37–52 (2003)
Okonek, Ch., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces. Progress in Mathematics, vol. 3. Birkhäuser, Boston (1980)
Ramanathan, A.: Moduli for principal bundles over algebraic curves I. Proc. Indian Acad. Sci. 106(3), 301–328 (1996)
Santos, J.P.: Framed holomorphic bundles on rational surfaces. J. Reine Angew. Math. 589, 129–158 (2005)
Reichstein, Z.: On the notion of essential dimension for algebraic groups. Transform. Groups 5(3), 265–304 (2000)
Schofield, A.: Birational classification of moduli spaces of vector bundles over \({\mathbb{P}}^2\). Indag. Math. (N.S.) 12(3), 433–448 (2001)
Simpson, C.T.: Moduli of representations of the fundamental group of a smooth projective variety I. Publ. Math. Inst. Hautes Études Sci. 79, 47–129 (1994)
Tikhomirov, A.A.: The main component of the moduli space of mathematical instanton vector bundles on \({\mathbb{P}}^3\). J. Math. Sci. (N. Y.) 86(5), 3004–3087 (1997)
Tikhomirov, A.S.: Moduli of mathematical instanton vector bundles with odd \(c_2\) on projective space. Izv. Math. 76(5), 991–1073 (2012)
Tikhomirov, A.S.: Moduli of mathematical instanton vector bundles with even \(c_2\) on projective space. Izv. Math. 77(6), 1195–1223 (2013)
Yoshioka, K.: Some notes on the moduli of stable sheaves on elliptic surfaces. Nagoya Math. J. 154, 73–102 (1999)