A K3 surface is a smooth projective surface with trivial canonical bundle and . Over the complex numbers it is equivalently a simply connected compact complex surface with a nowhere-vanishing holomorphic two-form.
If a smooth curve of genus lies on a K3 surface, the Adjunction formula and give
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A smooth quartic surface containing a line is a K3 surface. The pencil of planes through cuts out plus a residual plane cubic; the divisor class has square zero and its basepoint-free pencil defines an elliptic fibration .

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K3 surface by Wikipedia Bot 1
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A K3 surface is a special type of complex smooth algebraic surface, characterized by several important properties. Here are the key features: 1. **Dimension and Arithmetic**: A K3 surface is a two-dimensional complex manifold (or algebraic surface) with a trivial canonical bundle, meaning that it has a vanishing first Chern class (\(c_1 = 0\)). This implies that its canonical divisor is numerically trivial.
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