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For an elliptic curve $E$, projection ${\mathbb{P}}^1\times E\to {\mathbb{P}}^1$ is an elliptic fibration. Its canonical bundle is the pullback of $K_{{\mathbb{P}}^1}$, so it has no nonzero pluricanonical sections and its Kodaira dimension is $-\infty$.

The Iitaka dimension of a Cartier divisor $D$ is the maximum dimension of the images of the rational maps given by its nonempty multiples $|mD|$. It is $-\infty$ if all positive multiples have no sections. For an integral $n$-dimensional projective variety it lies between zero and $n$ when nonnegative. The Kodaira dimension of the variety is the special case $D=K_X$.

Mihnea Popa works on algebraic geometry, including the birational classification of varieties and the geometry of families. His Harvard course on Kodaira dimension documents the connection with positivity and Hodge theory.^[ref]

His Kodaira dimension of algebraic varieties by Mihnea Popa explains how sections of powers of the canonical bundle measure the growth of pluricanonical maps. For a smooth projective curve, the trichotomy between genus zero, genus one and higher genus appears as Kodaira dimension $-\infty$, zero and one. This gives a concrete entry into a classification invariant that remains meaningful in higher dimensions.^[ref]

A minimal algebraic surface is a smooth projective surface containing no exceptional curve of the first kind, namely no smooth rational curve of self-intersection $-1$. An abelian surface contains no rational curve because every morphism from ${\mathbb{P}}^1$ to an abelian variety is constant. It therefore has no $(-1)$-curve and is minimal.

For every $n\ge 2$, the Minimal Hirzebruch surface ${\mathbb{F}}_n$ is a rational minimal surface. These surfaces are pairwise nonisomorphic: the negative section is the unique irreducible curve of negative self-intersection and has square $-n$, so an isomorphism would recover $n$. Thus there are infinitely many nonisomorphic minimal rational surfaces.

A K3 surface is a smooth projective surface $X$ with $K_X\cong {\mathcal{O}}_X$ and $H^1(X,{\mathcal{O}}_X)=0$. For a smooth curve $C\subset X$ of geometric genus $g$, the adjunction formula givesand hence

An elliptic surface is a smooth projective surface with a morphism to a smooth curve whose generic fiber is a smooth genus-one curve. If $E$ is an elliptic curve, projectionis an elliptic fibration. Its canonical bundle is pulled back from $K_{{\mathbb{P}}^1}$, so every positive pluricanonical space vanishes and the Kodaira dimension is $-\infty$. This supplies the requested negative-Kodaira-dimension example.

For an elliptically fibered K3 surface, choose a smooth quartic $X\subseteq {\mathbb{P}}^3$ containing a line $L$. The canonical bundle of a smooth projective hypersurface formula makes $K_X$ trivial, and the standard cohomology sequence gives $H^1(X,{\mathcal{O}}_X)=0$, so $X$ is K3. The pencil of planes through $L$ cuts $X$ into $L$ plus a residual plane cubic. The residual linear system $|H-L|$ is basepoint-free, has square zero and defines a morphism $X\to {\mathbb{P}}^1$ whose generic fiber is a smooth plane cubic. This is the Elliptic K3 surface from a quartic containing a line.

A surface of general type is a smooth projective surface of Kodaira dimension two. Let $B\subseteq {\mathbb{P}}^2$ be a smooth plane curve of degree eight and letbe the degree-two cover branched along $B$. The branch-cover canonical-bundle formula givesThis divisor is ample, so $X$ is a double plane of general type and $\pi$ is the required finite morphism of degree two.

A smooth projective surface is of general type when its Kodaira dimension is two, equivalently when its canonical divisor is big.