# Different types of pencil of conics in the projective plane

Let $$\mathbb{K}$$ be an algebraically closed field. Let $$\mathcal{Q}$$ be the 6-dimensional vector space of the quadratic (or bilinear symmetric) form over $$\mathbb{K}^3$$, and $$\mathbb{P}(\mathcal{Q})$$ the 5-dimensional projective space associated. An element of $$\mathbb{P}(\mathcal{Q})$$ is a conic section over $$\mathbb{P}_2(\mathbb{K})=\mathbb{P}(\mathbb{K^3})$$.

A pencil of conics in $$\mathbb{P}(\mathcal{Q})$$ is a linear system of dimension $$1$$. My professor states that there are 5 different types of non-degenerate of pencil of conics:

• there exists 4 points such that the pencil is the pencil of the conics through them;
• there exists 3 points and a line through one of those points and not containig the other 2 points such that the pencil is the pencil of the conics through the 3 points and having the line as a tangent in the first point;
• there exists 2 points and 2 lines, where each line contains one of the 2 points and such that the pencil is the pencil of conics passing through the 2 points and in those points tangent to the 2 lines;
• the pencil is generated by a smooth conic ad a couple of lines such that the one of the 2 lines is tangent to the first conic and the intersection of the 2 lines and the conic is empty;
• the pencil is generated by a double line and a smooth conic, tangent to each other.

This result is obtained considering that in a pencil of conics, in the condition i described, there exist a smooth and a degenerate conic and treating the possible intersection cases using Bezout theorem. But I am not sure that the cases are all the possible one. In particular is it possible to reduce the following case to one of the cases quoted above: a smooth conic and a couple of lines, where the lines and the conic have a point in common, one line is tangent to the conic and the other intersects the conic at another point? This case is the one where the intersection multiplicity of the 2 conics is divided in one point, where it is 3, and another point where it is 1. The sum is 1 as Bezout says.

If you want you can give me a reference to the classification of the pencil of conics.

Thanks!!

• I don't understand the case you are refering to. Your second line should intersect your conic at 2 points (with multiplicity). – Carot Aug 23 at 16:33
• Yes, I cannot see the case I explained as one of the 5 mentioned above. I can try explaining it better, if this is the problem. – Alessandro Pecile Aug 23 at 19:28
• No I don't understand what case you are refering to. I don't think it is possible. – Carot Aug 23 at 21:06

It is true that there exist 5 types of non-degenerate pencils of conics (in case field $$\mathbb K$$ is algebraically closed). However, the fourth case you mentioned coincides with the second one. Indeed, considering case 2, denote the base points by $$a, b, c$$ and let $$l$$ be the mentioned line, $$a \in l$$, and let $$m$$ be the line passing through $$b,c$$. The pencil contains the conic $$l\cup m$$, since $$l$$ is tangent to this conic. Now, choosing any smooth conic $$C$$ in the pencil, one can see that $$C$$ does not pass through the point $$l\cap m$$, because it touches $$l$$ at point $$a$$, $$a\ne l\cap m$$. Thus the pencil is generated by conic $$C$$ and the couple of lines $$m\cup l$$. $$l$$ is tangent to $$C$$ and the intersection of the $$2$$ lines and the conic is empty, so one obtains exactly a pencil of the fourth type.