Download Algebraic Geometry and its Applications: Collections of by P. R. Masani (auth.), Chandrajit L. Bajaj (eds.) PDF
By P. R. Masani (auth.), Chandrajit L. Bajaj (eds.)
Algebraic Geometry and its Applications should be of curiosity not just to mathematicians but additionally to machine scientists engaged on visualization and comparable issues. The e-book relies on 32 invited papers provided at a convention in honor of Shreeram Abhyankar's sixtieth birthday, which was once held in June 1990 at Purdue collage and attended via many popular mathematicians (field medalists), computing device scientists and engineers. The keynote paper is through G. Birkhoff; different members contain such prime names in algebraic geometry as R. Hartshorne, J. Heintz, J.I. Igusa, D. Lazard, D. Mumford, and J.-P. Serre.
Read Online or Download Algebraic Geometry and its Applications: Collections of Papers from Shreeram S. Abhyankar’s 60th Birthday Conference PDF
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Extra info for Algebraic Geometry and its Applications: Collections of Papers from Shreeram S. Abhyankar’s 60th Birthday Conference
Example text
In [4] it was also shown that, in case of p = 7, the polynomial W(U) 26 Shreeram S. Abhyankar is reducible in k(z,w)[U] iff it factors into two irreducible polynomials of degree 3. We shall indirectly use this fact in investigating the reducibility of 1lJ(U). I say indirectly because, by trying out factors only of this type, we shall be able to find them! 4 Singularities of the Auxiliary Curve Thus we want to consider the auxiliary curve given by
P" ;2) replacing (z*, w* , U) we have (14) 'ljJ*(X, y, Z) = Zp-1 + ~(X -~(X + y)y[Zp-2 - YZp-3 + y 2Zp-4 + y)yp-3[y2 + XPl. + YP- 3Zl ° We know that if p = 7 then the genus of ¢* = is 2 and using this fact we want to factor 'ljJ*(~,'T},Z) in k(~,'T})[Zl into two monic factors of degree 3 in Z. 5 Parametrizing the Auxiliary Curve in Characteristic Seven Henceforth assuming the characteristic of the ground field k to be 7, we 34 Shreeram S. Abhyankar have the curve (1') ¢*(~, TJ) = 0 with ¢*(X, Y) = y3 + (X + 2)y2 - (2X + I)X7y - X8 of genus 2 with the three singularities; a double point at (~, TJ, 1) (-1, -1, 1) where the curve has a higher tacnode of index 4; a 6-fold point at (~, TJ, 1) = (0,1,0) where the curve has a higher tac-cusp of index 4; and a double point at (~, TJ, 1) = (0,0,1) where the curve has a higher tacnode of index 4.
P;l. 32 Shreeram S. Abhyankar neighbourhoods of (0, 1,0) which thus account for ~ double points in the genus formula. -] ----+ AO(W* + 2) = 1[1] ----+ K:o: z* = 00[2]--- ----+ AOl(W*) = _~[P;l] ----+ A02(W*) = w~: y* _ptl[p;l] = 00- + 1) = ~[1] 1]- ----+ A~(W* + 1) = Pt l [1] ----+ Aooi(W* + i) = 1[1] ----+ Aoo(W* ----+ K:oo : z* - 1 = O[p - for 2 ::::; i ::::; p - 1. -. l(z* + 1)[UP-2 - w*Up-3 + W*2Up-4 - ... + w*p- 3U] -1(z* + 1)[w*p-2 + z*P]. The three singularities (z*,w*,l) = (1,-1,1), (0,0,1), (0,1,0) of the curve <{>* = of degree p+ 2 account for p+l + (p_l)2 + pH = p2+3 double 2 2 2 2 ° P oints and hence its genus is (11) (p+l)p 2 (genus of <{>* p2+3 and thus 2' p-3 = 0) = -2- Square-root Parametrization of Plane Curves 33 and hence in particular: (8<» if p = 7 then (genus of * = 0) = 2.