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I'm a beginner in algebraic curves and as an exercise I'm playing with the Plücker's formula. I'm finding some inconsistency in these formulas and I would like to know where I'm wrong.

We know the dual curve of a dual curve of $F$ is the $F$ itself.

Now, let's $F$ be a non-singular cubic, then by Plücker's formula we have for the dual:

$d^{\vee}=d(d-1)$ and with $d=3$, we have $d^{\vee}=3\cdot 2=6$.

But if I use this formula again to find the degree of $F$ itself we have:

$(d^{\vee})^{\vee}=d^{\vee}(d^{\vee}-1)$ and with $d^{\vee}=6$, we have $(d^{\vee})^{\vee}=6\cdot 5=30 \neq 3=d$.

As we see, $d$ and $(d^{\vee})^{\vee}$ are not equal as we expected.

Anyone could clarifies this please?

Thanks in advance

user122342
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1 Answers1

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The dual of a plane cubic is not smooth. Rather, for each of the $9$-flexes of $F$, there is a cusp of $F^{\vee}$. The double dual $(F^{\vee})^{\vee}$ is the union of the original $F$, plus $9$ lines. Each of those lines occurs with multiplicity $3$, for reasons that aren't clear to me. (Hence $30=3+9 \times 3$.)

Remember that the dual curve should be thought of as living in the dual projective plane to the original curve. If $C$ in $\mathbb{P}$ has a singularity at $x \in \mathbb{P}$, then the dual curve $C^{\vee}$ in $\mathbb{P}^{\vee}$ will contain a line, which will be the line dual to $x$. A point on this line corresponds to a line in $\mathbb{P}$ through $x$. As a slogan "any line through a singularity is a tangent line".

For further discussion on the relation between the singularities of $C$ and the geometry of $C^{\vee}$, see Chapter 2.4 in Griffiths and Harris, Principles of Algebraic Geometry.

David E Speyer
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