Recent content by Mathmos6

Ok, I think I've got it now, thanks very much for the help!

Homework Statement I am trying to work out a solution to the following problem, where we are working in a field K complete with respect to a discrete valuation, with valuation ring \mathcal{O} and residue field k. Q: Let f(X) be a monic irreducible polynomial in K[X]. Show that if f(0) \in...

Ah yes of course, I was being stupid :smile: So then as you suggested we get out a quartic in the required variables, I'm just not sure what a "plane" quartic is? Thanks!

I'm a little confused about why you're doing this: rather than "rewriting" the old equation in terms of the images of X, Y, Z, wouldn't we take the map \varphi to "substitute in" YZ, XZ, XY for each instance of X, Y, Z respectively? For instance the first term of the equation would become...

Oh, of course, I'm sorry! I thought I wrote it down, obviously not :) \varphi: \mathbb{P}^2 - \to \mathbb{P}^2 is the rational map given by (X_0:X_1:X_2) \to (X_1 X_2: X_0 X_2: X_0 X_1) = (1/X_0: 1/X_1 : 1/X_2): I have shown already that this map is not regular at the points (1:0:0), (0:1:0)...

You've been such a brilliant help Micromass, thank you so much :) I do have one final question which has come up, if you don't mind. The final part of a problem I've been doing says: "let V \subset \mathbb{P}^2 be the plane curve given by the vanishing of the polynomial X_0^2X_1^3 + X_1^2 X_2^3...

Wonderful: that last post is fixed, I don't remember having to use itex the last time I used PF but perhaps that's just how long I've been away for! I concur with your blog, Hartshorne is boring and unintelligible, I don't like it at all but sadly it seems to be a bit of an industry standard, if...

Hi Micromass! Thanks for the quick response! I have primarily been using any online lecture notes I could find, but Hartshorne has come in handy once or twice too. I have seen the notion of singularity, yes: so essentially the tangent space has dimension 2 at the origin whereas the curve...

Homework Statement How would I go about showing that if X = \{(x,\,y) \in \mathbb{C}^2 | x^2 = y^3\} then X is birational but not isomorphic to the affine space \mathbb{A}^1[/tex]? I have found the obvious birational map, sending [itex](x,\,y) \to \frac{x}{y}, so I have shown the spaces...

My definition of a Hilbert space is standard, i.e. a real or complex inner product space which is complete under the norm defined by the inner product. The spectrum of an operator T is, for me, the set of points 'p' in the complex plane for which T-pI is not invertible (I the identity map). I...

Homework Statement Much as the title says, I need to construct an example of a linear operator on Hilbert space with empty spectrum. I can very easily construct an example with empty point-spectrum (e.g. the right-shift operator on l_2), but this has very far from empty spectrum. If I...

I apologise if it sounded like I was trying to avoid doing the work or learning from the problems myself but instead trying to find a book from which to copy them - indeed, precisely the reason why I didn't ask any specific questions on the problem sheets is because I want to do them myself in...

Hello everyone - I'm a third year student at Cambridge university, and I've recently started taking a course on Riemann surfaces along with a number of other pure courses this year. The problem is, the lecturer of the course is of a rather sub-par standard - whilst I don't doubt he's probably...

Ah of course, it makes perfect sense when you put it like that :) The argument is fairly simple once you spot it, I was definitely overcomplicating things - thankyou for being so patient!

I'm still not completely sure sorry, I'm obviously not getting this :( When you're looking at L over L∩K, the span of your elements will have to be smaller than or equal to the span of those elements over K, but I can't really see how to formulate this idea properly - sorry to keep asking! I'm...