# Projective subvarieties of blow-ups of affine varieties

Let $$X$$ be an integral affine scheme of finite type over a field $$k$$. Let $$Y\subset X$$ be an integral closed subscheme of codimension $$n>0$$. We blow up $$X$$ along $$Y$$ and get a $$k$$-scheme $$X'$$. Is it true that there is no closed immersion from a proper $$k$$-scheme of dimension $$\geq n$$ to $$X'$$?

No, this is not true. Take $$X = \mathbb{A}^4$$ and let $$Y = \{x_1x_3 - x_2^2 = x_1x_4 - x_2x_3 = x_2x_4 - x_3^2 = 0 \} \subset X$$ be the cone over a rational twisted cubic curve in $$\mathbb{P}^3$$. The codimension of $$Y$$ in $$X$$ is $$n = 2$$. On the other hand, it is easy to check that the fiber of the blowup $$X' \to X$$ over the origin (the vertex of the cone) is $$\mathbb{P}^2$$.
• what if we assume that $X$ and $Y$ are smooth? – rori Mar 8 at 12:25
• Since $X$ is affine, any proper subvariety of $X'$ is contained in a fiber of the blow up, which has dimension $\leq n-1$ if $X$ and $Y$ are smooth. – abx Mar 8 at 13:19