# How to easily introduce the projective tangent space?

Context

I'm a physicist writing a paper in which the notion of projective tangent space to a subspace of homogeneous polynomials is a cornerstone. However, the audience the paper is aimed at doesn't know anything about projective geometry or algebraic geometry. My goal is to introduce the definition of projective tangent space with as few elements as possible and as intuitively as possible. The best I made so far (which is not as simple as I would like) is the following

Let $$\mathbb{P}^n \equiv \mathbb{P}(\mathbb{C}[x,y]_N)$$. The affine tangent space to a polynomial $$\bar{P}$$ of a subset $$\widehat{X}$$ of $$\mathbb{C}[x,y]_N$$, denoted $$T_\bar{P}\widehat{X} \subset \mathbb{C}[x,y]_N$$, is the span of all polynomials in $$\mathbb{C}[x,y]_N$$ obtained as the derivative $$\alpha'(0)$$ of a smooth analytic parametrized curve $$\alpha : \mathbb{C} \rightarrow \widehat{X}$$ with $$\alpha(0) = \bar{P}$$. Let $$X$$ be the projective space of $$\widehat{X}$$, that is $$X=\mathbb{P}(\widehat{X})$$ and $$P\in \mathbb{P}^N$$ such that $$\bar{P}\in P$$. The projective tangent space to $$P$$ of $$X \subset \mathbb{P}^N$$ is $$\mathbb{P}(T_{\bar{P}}\widehat{X})$$

The difficulties with this definition are

• I don't like the definition of the cone over X I made, that is $$\widehat{X}$$ such that $$X=\mathbb{P}(\widehat{X})$$.
• The curve in the definition could be disappointing since the whole paper is about polynomials, not curves hose image is in $$\mathbb{C}[x,y]_N$$.
• This definition is too "mathematical". The way we use this notion of projective tangent space in the paper is different since we use a theorem that gives us precisely the shape of the elements of the projective tangent space. But I think that in order to introduce this theorem properly, it would be necessary to give the definition of the tangent space. The theorem is the following

Theorem. Let $$F = \prod_{i=1}^n L_i^{\lambda_i}$$ a smooth point of $$X_\lambda$$. The tangent space to $$X_\lambda$$ at F is given by $$T_FX_\lambda = \{h(x,y)\prod_{i=1}^n L_i^{\lambda_i -1} ~|~ h \in \mathbb{P}(\mathbb{C}[x,y]_n)\}.$$ where $$X_\lambda$$ is a projective variety.

Question

How would you introduce this notion of projective tangent space ? Would you make use of figures ?

• I am still looking for an answer :) Jun 21, 2023 at 8:37