Hyperplane sections of cubic threefolds

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X subset-of double-struck upper P Superscript 4"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo> ⊂ </mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">P</mml:mi> </mml:mrow> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">X\subset \mathbb {P}^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a smooth cubic hypersurface. Using the weak Lefschetz property for the Jacobian ring of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we prove that a general cubic surface is isomorphic to a hyperplane section of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .