  Let $X$ be a smooth irreducible projective variety of dimension at least 2
over 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 an
irreducible smooth subvariety of $X$. However, the precise location of the
smooth hyperplane section is not known. We show that for any $q\leq n+1$ closed
points in general position and any degree $a>1$, a general hypersurface $H$ of
degree $a$ passing through these $q$ points intersects $X$ with an irreducible
smooth codimension 1 subvariety on $X$. We also consider linear system of ample
divisors and give precise location of smooth elements in the system. Similar
result can be obtained for compact complex manifolds with holomorphic maps into
projective spaces.

Algebraic Geometry;Complex Variables