I am a bit confused about the definition of weak del pezzo surface. Can someone give an example that what kind of weak del pezzo surface is not a del pezzo surface?

A surface $S$ is del Pezzo if $-K_S$ is ample. It is weak del Pezzo if $-K_S$ is nef and big.

To get examples of (true) weak del Pezzos, remember that a del Pezzo of degree $d$ is the blowup of $\mathbf P^2$ in $9-d$ general points.

If $S$ is the blowup of $\mathbf P^2$ in points $p_1,\ldots,p_r$, then $-K_S=3H-E_1-\cdots-E_r$ (in the obvious notation).

So the trick is to choose the points so that $-K_S$ is nef and big, but has degree $0$ on some curve. For example, choose 6 points in $\mathbf P^2$ such that 3 of them lie on a line. Then on the blowup, $-K_S \cdot L=0$ where $L$ is the proper transform of the line. However, one can verify that $-K_S$ is still basepoint-free, hence nef, and has 4-dimensional space of sections, giving a birational map onto the image of $S$ in $\mathbf P^3$, hence is big.

The simplest example is the second Hirzebruch surface.

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