Asymptotic depth and connectedness in projective schemes

Abstract:

Let $I \subseteq \mathfrak {m}$ be an ideal of a local noetherian ring $\left ( {R,\mathfrak {m}} \right )$. Consider the exceptional fiber ${\pi ^{ - 1}}\left ( {V\left ( 1 \right )} \right )$ of the blowing-up morphism \[ \pi :\operatorname {Proj}\left ( {{ \oplus _{n \geq 0}}{I^n}} \right ) \to \operatorname {Spec}\left ( R \right )\] and the special fiber ${\pi ^{ - 1}}\left ( \mathfrak {m} \right )$. We show that the complement set \[ {\pi ^{ - 1}}\left ( {V\left ( I \right )} \right ) - {\pi ^{ - 1}}\left ( \mathfrak {m} \right )\] is highly connected if the asymptotic depth of the higher conormal modules ${I^n}/{I^{n + 1}}$ is large.