This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.

Definition. An integral projective scheme X over k is of general type if some desingularization of X is of general type. A reduced projective scheme X over k is of general type if every (reduced) irreducible component of $X$ is of general type. Finally, a projective scheme X is of general type if X_{red} is of general type.

My question is as follows.

Let S be an integral normal noetherian scheme of characteristic zero with function field $K=K(S)$. Let $X\to S$ be a projective flat morphism. Suppose that there is a closed point $s$ in $S$ with residue field $k(s) = k$ such that $X_s$ has an irreducible component which is of general type over $k$. Then the generic fibre $X_K$ of $X\to S$ is of general type over $K$.

Please note that I do not make any assumptions on the singularities of $X_s$, nor do I assume $X_s$ to be irreducible.

We may and do assume that $S$ is the spectrum of $\mathbb{C}[[t]]$.

A partial (positive) answer follows from deep theorems of Siu, Kawamata, and Nakayama on the constancy of plurigenera. But, as far as I know, these theorems require some conditions on the singularities of $X_s$ (e.g., canonical singularities). Can one reduce to this situation using semi-stable reduction, maybe?

This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.

Definition. An integral projective scheme X over k is of general type if some desingularization of X is of general type. A reduced projective scheme X over k is of general type if every (reduced) irreducible component of $X$ is of general type. Finally, a projective scheme X is of general type if X_{red} is of general type.

My question is as follows.

Let S be an integral normal noetherian scheme of characteristic zero with function field $K=K(S)$. Let $X\to S$ be a projective flat morphism. Suppose that there is a closed point $s$ in $S$ with residue field $k(s) = k$ such that $X_s$ has an irreducible component which is of general type over $k$. Is the generic fibre $X_K$ of $X\to S$ of general type over $K$?

Please note that I do not make any assumptions on the singularities of $X_s$, nor do I assume $X_s$ to be irreducible.

To answer this question, we may assume that $S$ is the spectrum of $\mathbb{C}[[t]]$.

A partial (positive) answer follows from deep theorems of Siu, Kawamata, and Nakayama on the constancy of plurigenera. But, as far as I know, these theorems require some conditions on the singularities of $X_s$ (e.g., canonical singularities). Can one reduce to this situation using semi-stable reduction, maybe?

This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.

Definition. An integral projective scheme X over k is of general type if some desingularization of X is of general type. A reduced projective scheme X over k is of general type if every (reduced) irreducible component of $X$ is of general type. Finally, a projective scheme X is of general type if X_{red} is of general type.

My question is as follows.

Let S be an integral variety over $k$ with function field $K=K(S)$. Let $X\to S$ be a projective flat morphism. Suppose that there is a closed point $s$ in $S(k)$ such that $X_s$ is of general type over $k$. Then the generic fibre $X_K$ of $X\to S$ is of general type over $K$.

Please note that I do not make any assumptions on the singularities of $X_s$, nor do I assume $X_s$ to be irreducible.

A partial (positive) answer follows from deep theorems of Siu, Kawamata, and Nakayama on the constancy of plurigenera. But, as far as I know, these theorems require some conditions on the singularities of $X_s$ (e.g., canonical singularities). Can one reduce to this situation using semi-stable reduction, maybe?

Probably, it suffices (and is more natural?) to assume only that at least one irreducible component of $X_s$ is of general type (instead of the entire fibre being of general type). This would at least allow one to reduce to the case that $X$ is smooth over $k$.

This question is about how varieties of general type over an algebraically closed field of characteristic zero $k$ behave under generization in families.

Definition. An integral projective scheme X over k is of general type if some desingularization of X is of general type. A reduced projective scheme X over k is of general type if every (reduced) irreducible component of $X$ is of general type. Finally, a projective scheme X is of general type if X_{red} is of general type.

My question is as follows.

Let S be an integral normal noetherian scheme of characteristic zero with function field $K=K(S)$. Let $X\to S$ be a projective flat morphism. Suppose that there is a closed point $s$ in $S$ with residue field $k(s) = k$ such that $X_s$ has an irreducible component which is of general type over $k$. Then the generic fibre $X_K$ of $X\to S$ is of general type over $K$.

Please note that I do not make any assumptions on the singularities of $X_s$, nor do I assume $X_s$ to be irreducible.

We may and do assume that $S$ is the spectrum of $\mathbb{C}[[t]]$.

A partial (positive) answer follows from deep theorems of Siu, Kawamata, and Nakayama on the constancy of plurigenera. But, as far as I know, these theorems require some conditions on the singularities of $X_s$ (e.g., canonical singularities). Can one reduce to this situation using semi-stable reduction, maybe?