Geodesic
𝔸1–connected components of ruled surfaces
Balwe, Chetan1 ; Sawant, Anand2
1 Department of Mathematical Sciences, Indian Institute of Science Education and Research, Mohali, India
2 School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India
Geometry & topology, Tome 26 (2022) no. 1, pp. 321-376.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

A conjecture of Morel asserts that the sheaf of 𝔸1–connected components of a space is 𝔸1–invariant. Using purely algebrogeometric methods, we determine the sheaf of 𝔸1–connected components of a smooth projective surface, which is birationally ruled over a curve of genus > 0. As a consequence, we show that Morel’s conjecture holds for all smooth projective surfaces over an algebraically closed field of characteristic 0.

Keywords: $\mathbb A^1$–connected components, ghost homotopies, ruled surfaces

Balwe, Chetan 1 ; Sawant, Anand 2

1 Department of Mathematical Sciences, Indian Institute of Science Education and Research, Mohali, India
2 School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India 
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Balwe, Chetan; Sawant, Anand. 𝔸1–connected components of ruled surfaces. Geometry & topology, Tome 26 (2022) no. 1, pp. 321-376. http://geodesic.mathdoc.fr/item/GT_2022_26_1_a7/

[1] M Artin, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math. 84 (1962) 485 | DOI

[2] A Asok, F Morel, Smooth varieties up to A1–homotopy and algebraic h–cobordisms, Adv. Math. 227 (2011) 1990 | DOI

[3] C Balwe, A Hogadi, A Sawant, A1–connected components of schemes, Adv. Math. 282 (2015) 335 | DOI

[4] C Balwe, A Sawant, R–equivalence and A1–connectedness in anisotropic groups, Int. Math. Res. Not. 2015 (2015) 11816 | DOI

[5] C Balwe, A Sawant, Naive A1–homotopies on ruled surfaces, preprint (2020)

[6] A Beauville, Complex algebraic surfaces, 34, Cambridge Univ. Press (1996) | DOI

[7] U Choudhury, Connectivity of motivic H–spaces, Algebr. Geom. Topol. 14 (2014) 37 | DOI

[8] U Görtz, T Wedhorn, Algebraic geometry, I : Schemes with examples and exercises, Vieweg and Teubner (2010) | DOI

[9] R L Graham, D E Knuth, O Patashnik, Concrete mathematics: a foundation for computer science, Addison-Wesley (1994)

[10] R Hartshorne, Algebraic geometry, 52, Springer (1977) | DOI

[11] J Kollár, Rational curves on algebraic varieties, 32, Springer (1996) | DOI

[12] F Morel, A1–algebraic topology over a field, 2052, Springer (2012) | DOI

[13] F Morel, V Voevodsky, A1–homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999) 45 | DOI

[14] A Sawant, Naive vs genuine A1–connectedness, from: "–theory" (editors V Srinivas, S K Roushon, R A Rao, A J Parameswaran, A Krishna), Hindustan (2018) 21