# How does the rational curve behave under the group action?

Let $$X$$ be $$\mathbb{P}^{2}$$(over $$\mathbb{C}$$) blowing up at 7 points in general position. Denote the points as $$p_{1}, p_{2}, …,p_{7}.$$ Denote the blowing up map as $$\pi_{1}: X\to \mathbb{P}^{2}$$.

We know there are 56 exceptional curves on $$X$$.

7 of them come from the total transform of $$p_{1}, p_{2},…,p_{7}$$, denote them as $$E_{i}$$

21 come from the strict transform of the lines passing through $$p_{i}p_{j}$$, denote them as $$E_{ij}$$

21 come from the strict transform of conics passing through 5 of the 7 points, denote them as $$D_{ij}$$, where the conic does not pass through $$p_{i}, p_{j}$$

7 come from cubics passing through all of 7 points and has double points at $$p_{i}$$, denote them as $$D_{i}$$.

It is well known that $$X$$ also could be realized double cover of $$\mathbb{P}^{2}$$ ramified at smooth quartic. Denote the double cover as $$\pi_{2}: X\to \mathbb{P}^{2}$$. The $$X$$ has an order 2 group action, denote the nontrivial action as $$\sigma$$. Then $$\pi_{2}: X/<\sigma>\to \mathbb{P}^{2}$$.

The 56 exceptional curves also come from 28 bitangents of the quartic.

It is not hard to show, actually $$\sigma(E_{i})=D_{i}, \sigma(D_{i})=\sigma(E_{i})$$. $$E_{i}, D_{i}$$ comes from the same bitangent line of the quartic.

Now my question is if we pick the general line $$l$$ on $$\mathbb{P}^{2}$$ which does not pass through any of 7 points, denote the preimage under the blowing up as $$l_{1}$$, what $$\sigma(l_{1})$$ looks like?

Actually $$l_{1}E_{1}=0, l_{1}D_{1}=3$$, thus $$\sigma(l_{1})E_{1}=3, \sigma(l_{1})D_{1}=0$$. However, I could not find a rational curve which intersect $$E_{1}$$ at 3 points and does not intersect with $$D_{1}$$.

## 1 Answer

The double covering $$\pi_2 \colon X \to \mathbb{P}^2$$ is the anticanonical morphism, therefore the image of $$l_1 = \pi_1^{-1}(l)$$ under $$\pi_2$$ is a cubic curve, hence $$\pi_2^{-1}(\pi_2(\pi_1^{-1}(l)))$$ is a triple anticanonical curve, and therefore $$\sigma(l_1) = \pi_2^{-1}(\pi_2(\pi_1^{-1}(l))) - l_1$$ corresponds to a curve of degree 8 in $$\mathbb{P}^2$$ that pass through each of the points $$p_i$$ with multiplicity 3.

• Thanks a lot! You answer is excellent. I know where I made a mistake. Basically, I thought rational singular curves on $\mathbb{P}^{2}$ are just nodal cubic and cusp cubic. However, this question show that a degree 8 curves passing through each $p_{i}$ at 3 times is also a rational curve. Am I right? By the way, could you show me an example that a singular rational curve on $\mathbb{P}^{2}$ which passing through a point for 3 times?
– Mike
Oct 6, 2022 at 12:57
• Take your favorite rational curve in $\mathbb{P}^4$ (e.g., normal quartic), choose a general triple of points, choose a general line in their linear span, and project out of this line to $\mathbb{P}^2$. Oct 6, 2022 at 13:14