Asymptotic depth and connectedness in projective schemes

Abstract

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