I would like to work through computing the blow-up of a particular curve along a subvariety consisting of just two points, both of which are ordinary double points.

Let $$F(X,Y,Z) = X^4 + Y^4 - X Y^2 Z - X^2 Z^2 + Y^2 Z^2 + Z^4 \in \mathbb{F}_3[X,Y,Z]$$ and let $C \subseteq \mathbb{P}^2 $ be the curve over $\mathbb{F}_3$ defined by $F(X,Y,Z) = 0$.

Now consider the subvariety of $C$ defined by $X^2 + Z^2 = Y = 0$, i.e. the two points $[2\alpha + 2, 0, 1], [\alpha + 1, 0, 1]$ where here $\alpha$ satisfies $\alpha^{2} + 2\alpha +2 = 0$.

I would like to compute the blow-up of $C$ along this subvariety. I know how to compute blow-ups at a single point but I am not sure how to go about computing this blow-up along two points.

EDIT: I forgot to emphasise that I am interested in blowing up a curve along a subvariety of just two points, where those points are (conjugate) ordinary double points.

  • $\begingroup$ You should be able to blow up at multiple points by blowing up each point separately. math.stackexchange.com/questions/97201/… $\endgroup$ – Ali Caglayan Apr 4 '18 at 10:24
  • $\begingroup$ Thank you. My problem is that I am trying to use this example to understand things about what the field of definition is for points on the blow-up which lie above singular points on the original curve C, and for this reason I believe I need to blow up the subvariety as defined above as opposed to blowing up each point individually. I want the polynomial equations whose zero set defines the (two) points to have coefficients from the field $F_3$, so I want to blow both points up at once as opposed to blowing them up one by one. $\endgroup$ – maddels Apr 4 '18 at 12:19

You can blowup any lci subvariety in the same way. In this particular case, consider the projective bundle $$ \mathbb{P}_{\mathbb{P}^2}(\mathcal{O} \oplus \mathcal{O}(-1)) $$ over $\mathbb{P}^2$. Let $U$ and $V$ be the coordinates corresponding to the line bundle summands in the projective bundle. Then the blowup of $\mathbb{P}^2$ is the subvariety in the projective bundle given by the equation $$ UY - V(X^2 + Z^2) = 0, $$ and the blowup of the curve is its strict transform in the blowup of the plane.

EDIT. Similarly to the blowup of a point, there are two charts: $U \ne 0$ and $V \ne 0$. On the first you can write $Y = v(X^2 + Z^2)$, substitute into the equation of the curve, and cancel the common factor; on the second you can write $X^2 + Z^2 = uY$, substitute, and cancel again.

  • $\begingroup$ 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. $\endgroup$ – maddels Apr 4 '18 at 12:04
  • $\begingroup$ 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. $\endgroup$ – maddels Apr 4 '18 at 12:13
  • $\begingroup$ 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. $\endgroup$ – maddels Apr 4 '18 at 12:26
  • $\begingroup$ @maddels: I added a short explanation. Is it more clear now? $\endgroup$ – Sasha Apr 4 '18 at 13:27
  • $\begingroup$ 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 $(X^2-(\alpha + \alpha^\prime) X Z + (\alpha \alpha^\prime) Z^2)$, $(Y^2-(\beta+ \beta^\prime) Y Z + (\beta \beta^\prime) Z^2)$. $\endgroup$ – maddels Apr 4 '18 at 13:51

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