Let $X, Y$ be irreducible projective varieties and $Y$ be smooth. Let $f:X \to Y$ be a flat projective morphism. Assume that a special fiber of $f$ is non-reduced i.e., there exists an irreducible component of the fiber which is of multiplicity greater than $1$. When can we say that the generic fiber or $X$ is non-reduced?

Let $X, Y$ be irreducible projective varieties and $Y$ be smooth. Let $f:X \to Y$ be a flat projective morphism. Assume that a special fiber of $f$ is non-reduced i.e., there exists an irreducible component of the fiber which is of multiplicity greater than $1$. When can we say that the generic fiber or $X$ is non-reduced?

EDIT Is there any criterion on $f$ or on $X, Y$ under which the generic fiber or $X$ is non-reduced provided it satisfies the above condition?

Let $X, Y$ be projective varieties and $Y$ be smooth. Let $f:X \to Y$ be a flat projective morphism. Assume that a special fiber of $f$ is non-reduced i.e., there exists an irreducible component of the fiber which is of multiplicity greater than $1$. When can we say that a generic fiber or $X$ is non-reduced?

Let $X, Y$ be irreducible projective varieties and $Y$ be smooth. Let $f:X \to Y$ be a flat projective morphism. Assume that a special fiber of $f$ is non-reduced i.e., there exists an irreducible component of the fiber which is of multiplicity greater than $1$. When can we say that the generic fiber or $X$ is non-reduced?