Bertini's theorem in family (Q2880703)

Let \(K\) be an arbitrary field and consider a (geometrically integral) projective variety \(X\) over \(K\), with \(\dim(X)\geq 2\). Bertini's theorem guarantees that the intersection of \(X\) with a general hypersurface of given degree, is geometrically integral. The author considers the subvarieties \(F^{int}_e\) of the space of hypersurfaces of degree \(e\), which parametrizes hypersurfaces whose intersection with \(X\) is \textit{not} geometrically integral. The target is a lower bound for the codimension of \(F^{int}_e\). The author is indeed able to find sharp lower bounds, which depend on \(e\), \(\dim(X)\) and on the existence of closed points of depth \(1\) in \(X\). As a consequence, the author provides a criterion for the existence of hypersurfaces of degree \(e\), which cut \textit{all} the elements of a given family \(\{X_t\}\) of (geometrically integral) varieties, in geometrically integral divisors.