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? enter image description here

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 1


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.

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    $\begingroup$ 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? $\endgroup$
    – Mike
    Oct 6, 2022 at 12:57
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    $\begingroup$ 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 $\mathbb{P}^2$. $\endgroup$
    – Sasha
    Oct 6, 2022 at 13:14

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