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Further evaluation of Wahl vanishing theorems for surface singularities in characteristic \(p\). (English) Zbl 1442.14114
Let \(X \longrightarrow \text{Spec}(R)\) be the minimal resolution of a rational double point defined over an algebraically closed field. Let \(E\) be the exceptional divisor and \(S_X=\Theta_X(-\text{log }E)\) the sheaf of logarithmic derivations. It is proved that the canonical morphism \(H^1(S_X) \longrightarrow H^1(X \setminus E,S_X)\) is an inclusion. The dimension of \(H^1_E(S_X\otimes \mathcal O_X(E))\) is computed. It is proved that \(H^0(X\setminus E,\Theta_X)/H^0(X,\Theta_X)\) is isomorphic to \(H^1_E(S_X)\) and the dimension is computed.
14J17 Singularities of surfaces or higher-dimensional varieties
32S25 Complex surface and hypersurface singularities
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[1] M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129-136. · Zbl 0142.18602
[2] M. Artin, Algebraic construction of Brieskorn’s resolutions, J. Algebra 29 (1974), 330-348. · Zbl 0292.14013
[3] M. Artin, Coverings of the rational double points in characteristic \(p (W\). L. Jr. Baily, T. Shioda eds.), Complex analysis and algebraic geometry, pp. 11-22, Cambridge Univ. Press, Cambridge, 1977.
[4] D. M. Burns Jr. and J. M. Wahl, Local contributions to global deformations of surfaces, Invent. Math. 26 (1974), 67-88. · Zbl 0288.14010
[5] R. Fedder, \(F\)-purity and rational singularity, Trans. Amer. Math. Soc. 278 (1983), 461-480. · Zbl 0519.13017
[6] M. Hirokado, Canonical singularities of dimension three in characteristic \(2\) which do not follow Reid’s rules, Kyoto J. Math. (to appear).
[7] M. Hirokado, H. Ito and N. Saito, Three dimensional canonical singularities in codimension two in positive characteristic, J. Algebra 373 (2013), 207-222. · Zbl 1271.14047
[8] F. Huikeshoven, On the versal resolutions of deformations of rational double points, Invent. Math. 20 (1973), 15-33. · Zbl 0268.32010
[9] S. Ishii and A. J. Reguera, Singularities in arbitrary characteristic via jet schemes (L. Ji ed.), Hodge theory and \(L^2\)-analysis, Adv. Lectures Math., 39, pp. 419-449, International Press, Boston, 2017. · Zbl 1379.14022
[10] C. Liedtke and M. Satriano, On the birational nature of lifting, Adv. Math. 254 (2014), 118-137. · Zbl 1319.14015
[11] H. Matsumura, Commutative ring theory, Cambridge Univ. Press, Cambridge, 1986. · Zbl 0603.13001
[12] M. Reid, Canonical \(3\)-folds (A. Beauville ed.), Journées de Géométrie Algébrique d’Angers 1979, pp. 273-310, Sijthoff & Noordhoff, 1980.
[13] K. Sato and S. Takagi, General hyperplane sections of threefolds in positive characteristic, J. Inst. Math. Jussieu. (to appear).
[14] N. I. Shepherd-Barron, On simple groups and simple singularities, Israel J. Math. 123 (2001), 179-188. · Zbl 1007.14001
[15] P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Math., 815, Springer, Berlin-Heidelberg-New York, 1980. · Zbl 0441.14002
[16] T. A. Springer and R. Steinberg, Conjugacy classes (Borel et al. eds.), Seminar on algebraic groups and related finite groups, Lecture Notes in Math., 131, pp. 167-266, Springer, Berlin-Heidelberg-New York, 1970.
[17] G. N. Tjurina, Absolute isolatedness of rational singularities and triple rational points, Funct. Anal. Appl. 2 (1968), 324-333.
[18] J. M. Wahl, Vanishing theorems for resolutions of surface singularities, Invent. Math. 31 (1975), 17-41. · Zbl 0314.14010
[19] J. M. Wahl, Equisingular deformations of normal surface singularities, I, Ann. of Math. 104 (1976), 325-356. · Zbl 0358.14007
[20] J. M. Wahl, The number of equisingular moduli of a rational surface singularity, Methods Appl. Anal. 24 (2017), 125-154. · Zbl 1391.14009
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