Exercise 1.6.1 Shafarevich Basic Algebraic Geometry 1

Question

I have been struggling with the following question for a while from section 1.6 of Shafarevich's Basic Algebraic Geometry I:

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 $\mathbb{P}^n-L$. Join $y$ to all points $x\in X$ by lines, and denote by $Y$ the set of points lying on all these lines, that is, the cone over $X$ with vertex $y$. Prove that $Y$ is an irreducible projective variety and $\dim Y = \dim X + 1$.

This seems like it would have a constructive answer, i.e. given some $k$ equations in $n$ variables that define $X\subset L$, it seems like I could construct $k$ equations that define $Y$ in $\mathbb{P}^n$, which would obviously give the dimension result. However I cannot seem to figure out how such a construction would work.

Is my approach valid, or should I be approaching this a different way?

Answer 1

This is not really the way one should go about this. For one, the connection between the number of polynomials needed to define a subvariety and it's dimension is not so strong: clearly any subvariety $X\subset \Bbb P^n$ of codimension $c$ is cut out by at least $c$ polynomials, but it need not be the case that $I(X)$ is generated by $c$ polynomials. For instance, the twisted cubic curve in $\Bbb P^3$ has an ideal which cannot be generated by less than three elements, but is codimension two.

Here are some better approaches.

• Use that dimension is a birational invariant. Try to find a birational equivalence of $Y$ with a variety which is obviously of dimension $\dim X+1$.

• Cut with a hyperplane and induct. If $H$ is a hyperplane not containing $Y$, then $\dim H\cap Y = \dim Y-1$. Pick a hyperplane which passes through $P$ but doesn't contain $X$, and observe that $J(X,P)\cap H=J(X\cap H,P)$.

• You can also pursue Tabes Bridges' solution from the comments: take a chain of irreducible closed subvarieties witnessing the dimension of $X$, and take their joins.

I leave the finer details to you - please write me a comment if you run in to trouble.