Timeline for An explicit computation of the blow-up of curve over $\mathbb{F}_3$ at two points
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Apr 4, 2018 at 14:58 | comment | added | Sasha | I meant that the intersection might have two irreducible components, one of which is contained in the preimage of the blowup center. | |
| Apr 4, 2018 at 14:37 | comment | added | maddels | What do you mean by "up to an irreducible component that one should neglect?". What I would like is to be able to figure out what values $u$ should take when we are over the singular point on the original curve, i.e. by "substituting" (or something like substituting), and then taking $(X,Y,Z) = (a,b,1)$, obtain a quadratic $q(u)$ in $u$, whose roots are say $\gamma, \gamma^\prime$. So that then the points on the blow-up lying over $(a:b:1)$ are $((a:b:1), \gamma)$, $((a:b:1), \gamma^\prime)$. | |
| Apr 4, 2018 at 14:26 | comment | added | Sasha | @maddels: I shouldn't say "substitute $X^2+Z^2=uY$". This should be understood as that the curve is given by the two equations $X^2+Z^2=uY$ and $F(X,Y,Z) = 0$ (up to an irreducible component that one should neglect) in the product $\mathbb{P}^2 \times \mathbb{A}^1$. | |
| Apr 4, 2018 at 14:15 | comment | added | maddels | Thank you for your responses so far, they have been helpful! Sorry for all the extra questions. | |
| Apr 4, 2018 at 14:14 | comment | added | maddels | It seems I kind kind of substitute $X^2 + Z^{2} = uY$ into the original equation by $$F(u*X, X^2 + Z^{2}, u*Z)/(u^2(X^2 + Z^{2})^2) = 0$$ but I'm not sure how to do the more general thing I mention above, i.e. how to deal with the situation where we have $$V(X^2-(\alpha + \alpha^\prime) X Z + (\alpha \alpha^\prime) Z^2) - U(Y^2-(\beta+ \beta^\prime) Y Z + (\beta \beta^\prime) Z^2) = 0$$ | |
| Apr 4, 2018 at 14:03 | comment | added | maddels | For an example of this more general case I am concerned with, consider $$F(X,Y,Z) = -X^4 + X^3 Y - X^2 Y^2 - X Y^3 + X^3 Z + Y^3 Z - X^2 Z^2 + X Y Z^2 - Y^2 Z^2 + X Z^3 + Z^4$$ where $F \in \mathbb{F}_{3}[X,Y,Z]$. The singular points here are $P = (2\alpha + 2 : \alpha + 2 : 1), P^{\prime} = (\alpha + 1 : 2 \alpha : 1)$ where $\alpha^{2} + 2 \alpha + 2 = 0$. So the defining equations for the subvariety should be $$X^2 + Z^{2} = Y^{2} - 2YZ+ 2 Z^{2} = 0$$ Then I'd have $X^{2} + Z^2 = u( Y^{2} - 2YZ+ 2 Z^{2})$ for the $V \neq 0$ part, and $Y^2- 2YZ+ 2 Z^2 = v(X^{2} + Z^{2})$ for $U \neq 0$ | |
| Apr 4, 2018 at 13:52 | comment | added | maddels | In this more general case, how do I go about substituting into the original curve? | |
| Apr 4, 2018 at 13:51 | comment | added | maddels | Thank you, yes, that helps. Actually, would you be able to tell me how you would go about substituting $X^2 + Z^2 = uY$ into the equation of the curve? In addition, I would like to know about the more general situation where the two conjugate singular points on the curve are nodes, say of the form $(\alpha : \beta : 1), (\alpha^\prime : \beta^\prime : 1)$, where both $\alpha$ and $\beta$ may lie in $\mathbb{F}_{q^2}$, and thus we have two quadratics $(X^2-(\alpha + \alpha^\prime) X Z + (\alpha \alpha^\prime) Z^2)$, $(Y^2-(\beta+ \beta^\prime) Y Z + (\beta \beta^\prime) Z^2)$. | |
| Apr 4, 2018 at 13:27 | comment | added | Sasha | @maddels: I added a short explanation. Is it more clear now? | |
| Apr 4, 2018 at 13:26 | history | edited | Sasha | CC BY-SA 3.0 |
added 288 characters in body
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| Apr 4, 2018 at 12:26 | comment | added | maddels | Sorry, to clarify, what I am talking about above assumes the point (0:0:1) is a node (ordinary double point) on a curve C defined by $F(X,Y,Z) = 0$ where $F(X,Y,Z) \in \mathbb{F}_q[X,Y,Z]$. Similarly, the situation I posted about is a curve and I am trying to blow-up the curve along two (conjugate) nodes. | |
| Apr 4, 2018 at 12:13 | comment | added | maddels | When I say I don't know much algebraic geometry, I mean that I do not currently know about things like projective bundles, and I am just learning commutative algebra now from Atiyah-Macdonald. I am going off a rough understanding of the definition of blow-up from Harris' Algebraic Geometry in dealing with this problem. | |
| Apr 4, 2018 at 12:04 | comment | added | maddels | Thanks. I am an undergraduate and do not currently know much algebraic geometry, but will try to learn it. Would you be able to tell me how to go about getting the strict transform? I know that when blowing up at just one point, say at $(0: 0: 1)$, we can compute affine charts of the blow-up. Then in the affine chart where $V \neq 0$ the defining equations are: $$X - uY = 0$$ $$\frac{F(uY, Y, X)}{Y^2} = 0$$ I would like to know what the analogous equations are for my case of two points. | |
| Apr 4, 2018 at 11:10 | history | answered | Sasha | CC BY-SA 3.0 |