Elliptic K3 surface from a quartic containing a line
= Elliptic K3 surface from a quartic containing a line
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A smooth quartic surface $X\subseteq\mathbb P^3$ containing a line $L$ is a <K3 surface>. The pencil of planes through $L$ cuts out $L$ plus a residual plane cubic; the divisor class $H-L$ has square zero and its basepoint-free pencil defines an <elliptic surface>[elliptic fibration] $X\to\mathbb P^1$.