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Let me briefly expand on Jason's comment.

Actually, "formal scheme" is the right word here.

For any K3 surface $X$ over an algebraically closed field $k$ of any characteristic one has $$h^0(X, T_X)=0, \quad h^1(X, T_X)=20, \quad \, h^2(X, T_X)=0,$$ so the functor of Artin rings $$F \colon (\textbf{Art}) \to (\textbf{Sets})$$ describing the first-order deformations of $X$ is pro-representable and unobstructed, hence it is represented by a complete, regular local ring $A$ of dimension $20$. More precisely, $$A \cong k[[x_1, \ldots, x_{20}]],$$ where $x_1, \ldots, x_{20}$ are indeterminates.

Thus there is a formal universal deformation $\widehat{\mathcal{X}} \to \textrm{Specf} (A)$ of $X$; however, such a deformation is not effective, in the sense that there exists no algebraic deformation $\mathcal{X} \to \textrm{Spec}(A)$, where $ \mathcal X$ is a scheme, such that $\widehat{\mathcal{X}}$ is the formal completion of $\mathcal{X}$ along the closed fibre $X$. In fact, a straightforward cup product computation shows that algebraic family of K3 surfaces with injective Kodaira-Spencer map at any point has dimension $\leq 19$.

Deligne showed that any K3 surface over $k$ can be lifted to $\mathbb{C}$. So there is an analogy with the complex analytic theory, where the deformation space, as a complex manifold, has dimension $20$, but the algebraic K3 surfaces form a (countable union of) $19$-dimensional subfamilies.

More details on this topic can be found in the books Deformations of algebraic schemes (Sernesi) and Deformation theory (Hartshorne).

Let me briefly expand on Jason's comment.

Actually, "formal scheme" is the right word here.

For any K3 surface $X$ over an algebraically closed field $k$ of any characteristic one has $$h^0(X, T_X)=0, \quad h^1(X, T_X)=20, \quad \, h^2(X, T_X)=0,$$ so the functor of Artin rings $$F \colon (\textbf{Art}) \to (\textbf{Sets})$$ describing the first-order deformations of $X$ is pro-representable and unobstructed, hence it is represented by a complete, regular local ring $A$ of dimension $20$. More precisely, $$A \cong k[[x_1, \ldots, x_{20}]],$$ where $x_1, \ldots, x_{20}$ are indeterminates.

Thus there is a formal universal deformation $\widehat{\mathcal{X}} \to \textrm{Specf} (A)$ of $X$; however, such a deformation is not effective, in the sense that there exists no algebraic deformation $\mathcal{X} \to \textrm{Spec}(A)$, where $ \mathcal X$ is a scheme, such that $\widehat{\mathcal{X}}$ is the formal completion of $\mathcal{X}$ along the closed fibre $X$. In fact, a straightforward cup product computation shows that any algebraic family of K3 surfaces with injective Kodaira-Spencer map at any point has dimension $\leq 19$.

Deligne showed that any K3 surface over $k$ can be lifted to $\mathbb{C}$. So there is an analogy with the complex analytic theory, where the deformation space, as a complex manifold, has dimension $20$, but the algebraic K3 surfaces form a (countable union of) $19$-dimensional subfamilies.

More details on this topic can be found in the books Deformations of algebraic schemes (Sernesi) and Deformation theory (Hartshorne).

Let me briefly expand on Jason's comment.

Actually, "formal scheme" is the right word here.

For any K3 surface $X$ over an algebraically closed field $k$ of any characteristic one has $$h^0(X, T_X)=0, \quad h^1(X, T_X)=20, \quad \, h^2(X, T_X)=0,$$ so the functor of Artin rings $$F \colon (\textbf{Art}) \to (\textbf{Sets})$$ describing the first-order deformations of $X$ is pro-representable and unobstructed, hence it is represented by a complete, regular local ring $A$ of dimension $20$. More precisely, $$A \cong k[[x_1, \ldots, x_{20}]],$$ where $x_1, \ldots, x_{20}$ are indeterminates.

Thus there is a formal universal deformation $\widehat{\mathcal{X}} \to \textrm{Specf} (A)$ of $X$; however, such a deformation is not effective, in the sense that there exists no algebraic deformation $\mathcal{X} \to \textrm{Spec}(A)$, where $X$ is a scheme, such that $\widehat{\mathcal{X}}$ is the formal completion of $\mathcal{X}$ along the closed fibre $X$. In fact, a straightforward cup product computation shows that algebraic family of K3 surfaces with injective Kodaira-Spencer map at any point has dimension $\leq 19$.

Deligne showed that any K3 surface over $k$ can be lifted to $\mathbb{C}$. So there is an analogy with the complex analytic theory, where the deformation space, as a complex manifold, has dimension $20$, but the algebraic K3 surfaces form a (countable union of) $19$-dimensional subfamilies.

More details on this topic can be found in the books Deformations of algebraic schemes (Sernesi) and Deformation theory (Hartshorne).

Let me briefly expand on Jason's comment.

Actually, "formal scheme" is the right word here.

For any K3 surface $X$ over an algebraically closed field $k$ of any characteristic one has $$h^0(X, T_X)=0, \quad h^1(X, T_X)=20, \quad \, h^2(X, T_X)=0,$$ so the functor of Artin rings $$F \colon (\textbf{Art}) \to (\textbf{Sets})$$ describing the first-order deformations of $X$ is pro-representable and unobstructed, hence it is represented by a complete, regular local ring $A$ of dimension $20$. More precisely, $$A \cong k[[x_1, \ldots, x_{20}]],$$ where $x_1, \ldots, x_{20}$ are indeterminates.

Thus there is a formal universal deformation $\widehat{\mathcal{X}} \to \textrm{Specf} (A)$ of $X$; however, such a deformation is not effective, in the sense that there exists no algebraic deformation $\mathcal{X} \to \textrm{Spec}(A)$, where $ \mathcal X$ is a scheme, such that $\widehat{\mathcal{X}}$ is the formal completion of $\mathcal{X}$ along the closed fibre $X$. In fact, a straightforward cup product computation shows that algebraic family of K3 surfaces with injective Kodaira-Spencer map at any point has dimension $\leq 19$.

Deligne showed that any K3 surface over $k$ can be lifted to $\mathbb{C}$. So there is an analogy with the complex analytic theory, where the deformation space, as a complex manifold, has dimension $20$, but the algebraic K3 surfaces form a (countable union of) $19$-dimensional subfamilies.

More details on this topic can be found in the books Deformations of algebraic schemes (Sernesi) and Deformation theory (Hartshorne).

Let me briefly expand on Jason's comment.

Actually, "formal scheme" is the right word here.

For any K3 surface $X$ over an algebraically closed field $k$ of any characteristic one has $$h^0(X, T_X)=0, \quad h^1(X, T_X)=20, \quad \, h^2(X, T_X)=0,$$ so the functor of Artin rings $$F \colon (\textbf{Art}) \to (\textbf{Sets})$$ describing the first-order deformations of $X$ is prorepresentable and unobstructed, hence it is represented by a complete, regular local ring $A$ of dimension $20$. More precisely, $$A \cong k[[x_1, \ldots, x_{20}]],$$ where $x_1, \ldots, x_{20}$ are indeterminates.

Thus there is a formal universal deformation $\widehat{\mathcal{X}} \to \textrm{Specf} (A)$ of $X$; however, such a deformation is not effective, in the sense that there exists no algebraic deformation $\mathcal{X} \to \textrm{Spec}(A)$, where $X$ is a scheme, such that $\widehat{\mathcal{X}}$ is the formal completion of $\mathcal{X}$ along the closed fibre $X$. In fact, a straightforward cup product computation shows that algebraic family of K3 surfaces with injective Kodaira-Spencer map at any point has dimension $\leq 19$.

Deligne showed that any K3 surface over $k$ can be lifted to $\mathbb{C}$. So there is an analogy with the complex analytic theory, where the deformation space, as a complex manifold, has dimension $20$, but the algebraic K3 surfaces form a (countable union of) $19$-dimensional subfamilies.

More details on this topic can be found in the books Deformations of algebraic schemes (Sernesi) and Deformation theory (Hartshorne).

Let me briefly expand on Jason's comment.

Actually, "formal scheme" is the right word here.

For any K3 surface $X$ over an algebraically closed field $k$ of any characteristic one has $$h^0(X, T_X)=0, \quad h^1(X, T_X)=20, \quad \, h^2(X, T_X)=0,$$ so the functor of Artin rings $$F \colon (\textbf{Art}) \to (\textbf{Sets})$$ describing the first-order deformations of $X$ is pro-representable and unobstructed, hence it is represented by a complete, regular local ring $A$ of dimension $20$. More precisely, $$A \cong k[[x_1, \ldots, x_{20}]],$$ where $x_1, \ldots, x_{20}$ are indeterminates.

Thus there is a formal universal deformation $\widehat{\mathcal{X}} \to \textrm{Specf} (A)$ of $X$; however, such a deformation is not effective, in the sense that there exists no algebraic deformation $\mathcal{X} \to \textrm{Spec}(A)$, where $X$ is a scheme, such that $\widehat{\mathcal{X}}$ is the formal completion of $\mathcal{X}$ along the closed fibre $X$. In fact, a straightforward cup product computation shows that algebraic family of K3 surfaces with injective Kodaira-Spencer map at any point has dimension $\leq 19$.

Deligne showed that any K3 surface over $k$ can be lifted to $\mathbb{C}$. So there is an analogy with the complex analytic theory, where the deformation space, as a complex manifold, has dimension $20$, but the algebraic K3 surfaces form a (countable union of) $19$-dimensional subfamilies.

More details on this topic can be found in the books Deformations of algebraic schemes (Sernesi) and Deformation theory (Hartshorne).