There is an exercise that I am thinking about.
Let $ L \subset \mathbb{P}^{n} $ be an $ (n-1)$-dimensional linear subspace, $ X \subset L $ an irreducible closed variety and $ y $ a point in $ \mathbb{P}^{n} \backslash L. $ Say we draw lines from $ y $ to all points $ x \in X, $ and denote the set of points on these lines by $ Y. $ So $ Y $ is the cone over $ X $ with with vertex $ y. $ The exercise requires that one show that $ Y $ is van irreducible projective variety and $ \text{dim}Y = \text{dim}X + 1. $
I don't know how to even begin to show this from scratch on my own, but I have noticed a related question .i.e. The join of disjoint varieties in disjoint linear spaces has dimension $\dim X +\dim Y+1$.
Our cone is the join $ J(X,y), $ that is the set of lines from $ y $ to all $ x \in X. $ According to the book by Harris(page 148):
Theorem: Let $ X,Y \subset \mathbb{P}^{n} $ be two disjoint varieties. The join $ J(X,Y) $ will have dimension exactly $ d = \text{dim}X + \text{dim}Y + 1 $ whenever $ d \leq n. $
So by this, $$ \text{dim}J(X,y) = \text{dim}Y = \text{dim}X + 1. $$
Is there another way to establish this result? On the subject of the irreducibility of $ J(X,y), $ I don't know how to get started. Indeed, it's not even clear to me that it is a projective variety in the first place.