What exactly does it mean by pencil in this context?

I wanted to clarify some terminology. I am writing a project on elliptic curves and a document I am following mentions the following:

The cubics through ${P_1,...,P_8}$ form a pencil, $$\lambda_1\Gamma_1 + \lambda_2\Gamma_2$$

am I correct in thinking this is essentially saying that we can find two linearly independent cubics ${\Gamma_1,\Gamma_2}$ which pass through ${P_1,...,P_8}$ such that any other cubic passing through ${P_1,...,P_8}$ can be written as a linear combination of ${\Gamma_1,\Gamma_2}$?

Solution 1:

You are correct. In fact $\Gamma_1, \Gamma_2$ can be any two cubics passing through those points.

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