Bertini Type Theorems

TL;DR

This paper extends Bertini's Theorem by showing that for a smooth projective variety, general hypersurfaces passing through specified points intersect the variety smoothly and irreducibly, with precise control over the location of these intersections.

Contribution

It provides a detailed analysis of the location of smooth hyperplane sections and extends the classical Bertini theorem to hypersurfaces passing through fixed points and to complex manifolds.

Findings

General hypersurfaces passing through fixed points intersect the variety smoothly and irreducibly.

Precise location of smooth elements in linear systems of ample divisors is determined.

Results extend to compact complex manifolds with holomorphic maps into projective spaces.

Abstract

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.