Hi, in addition to your wonderful lecture i highly recommend the papaer "resolution of singularities : an introductio," by Marc spivakovsky, which gives an historical lecture of resolution of singularities by different methods (including the most important one : blowing-ups). Your examples plus his more theoretical point of view are really working together ! Thanks for your great lessons
Two more excelent (imho) related papers (depending on what one is interested in) are: Hauser - Seven short stories on blowups and resolutions (homepage.univie.ac.at/herwig.hauser/Publications/ggt05-hauser.pdf) Hauser - The Hironaka theorem on resolution of singularities ... (homepage.univie.ac.at/herwig.hauser/Publications/hauser%20hironaka%20thm%20bams.pdf)
The blow up at (0,0) of 2-dimensional affine plane is the set of points ( (x,y); [X:Y]) satisfying xY = yX, which is not orientable. Note that real projective line is orientable, that is there is a nowhere vanishing vector field.In fact, let s, t (=1/s) be local coordinates of projective line, then f(s)d/ds = f(1/t)(-1/t^2)d/dt. Thus, to get good vector fields, we can assume that there are constants a,b,c so that f(s) = a + bs + cs^2, from which f(1/t)(-1/t^2) = -at^2-^bt -c. Both of f(s)d/ds and f(1/t)(-1/t^2) does not have zeros on real numbers if b^2-4ac < 0. Taking a,b,c such that b^2-4ac < 0, we can get a nowhere vanishing vector field on the projective line. The correspondence from product of two copies of P^1 into P^2 is defined by sending (x:y) and (X:Y) to (xX:yX:xY). This is not a map, because the point of x = X = 0 is sent to (0:0:0) which is not a point of P^2. To define this map at the point x = X = 0, we consider the blow up at x = X = 0, that is ( (0:1), (0:1) ), which is done by introducing new local coordinate t of projective line and relation x = tX. The new correspondence from blow up to P^2 is (xX:yX:xY) = (tX^2: yX: tXY) = (tX: y: tY), which is well defined at x = X = 0. In fact, at x = X =0 we get (xX:yX:xY) = (tX^2: yX: tXY) = (tX: y: tY) = (0: y: tY), which does not vanish since y is not zero. Humn, I am not sure. The Blow up of P^1 times P^1 is a submanifold of the products of three copies of P^1. 10:14 What the inverse image of (0:0:1) ? If (tX: y: tY) = (0:0:1), then y = 0, so x = tX is nonzero, which is contradicts the first component of (tX: y: tY) = (0:0:1). Using another new local coordinate s, that is sx = X instead of x = tX, we get (xX:yX:xY) = (sx:ys:Y). In this coordinate, the inverse image of (0:0:1) is the set of points ((x:y), (X:Y). (s:t) ) = ( (x:y), (0:1) , (0:1)) for any (x:y). So, the inverse image is a projective line and we can regard this as an exceptional curve, I guess. 10:14 I am not sure .. the blow up of P^1 times P^1 is isomorphic to the blow up of P^1 at the two points (0:0:1) and (0:1:0)?? Geometric meaning of 17:11: I do not understand well. Blowing up at ideals of k[x,y]. Take an ideal of k[x,y] generated by n polynomials G1(x,y),….Gn(x,y). Blowing up along an ideal (G1(x,y),…,Gn(x,y) ) is defined as the closure of the points ( (x,y), [ G1(x,y):…:Gn(x,y) ] ) in A^2 times n copies of projective line, where (x,y) moves in complements of zeros of the ideal (G1(x,y),…,Gn(x,y) ). Blowing up of a point is the special case that G1(x,y) = x and G2(x,y) = y and if we denote the homogenous coordinate by [X:Y], we get [G1:G2] = [x:y] = [X:Y], which is equivalent to xY = yX. This relation appeared early. If we take G1(x,y) = x^2 and G2(x,y) = y^2, then [G1:G2] = [x^2:y^2] = [X:Y] which is equivalent to x^2Y = y^2X. Taking X = 1, we get the local coordinate representation x^2Y = y^2 which has singularity. So, the blowing up of nonreduced ideal (x^2, y^2) has a singularity.
