# Bertini Type Theorems - Mathematics > Algebraic Geometry

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Abstract: Let $X$ be a smooth irreducible projective variety of dimension at least 2over an algebraically closed field of characteristic 0 in the projective space${\mathbb{P}}^n$.Bertini-s Theorem states that a general hyperplane $H$ intersects $X$ with anirreducible smooth subvariety of $X$. However, the precise location of thesmooth hyperplane section is not known. We show that for any $q\leq n+1$ closedpoints in general position and any degree $a>1$, a general hypersurface $H$ ofdegree $a$ passing through these $q$ points intersects $X$ with an irreduciblesmooth codimension 1 subvariety on $X$. We also consider linear system of ampledivisors and give precise location of smooth elements in the system. Similarresult can be obtained for compact complex manifolds with holomorphic maps intoprojective spaces.

Author: ** Jing Zhang**

Source: https://arxiv.org/