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Given two smooth algebraic curves of degrees $d$ and $e$ in the complex projective plane, Bezout’s theorem is a classical result asserting that the intersection consists of either infinitely many or exactly $d\cdot e$ points, counted with multiplicities. Generalizing this formula to intersections in higher-dimensional projective space was a task that fuelled much of 20th century algebraic geometry, eventually culminating in J.-P. Serre’s solution in 1958.

We will review some of this history before presenting a 21st century point of view on Bézout’s theorem. Specifically, we will see that the modern language of derived algebraic geometry allows us to give a simple definition of intersection multiplicities, which unlike Serre’s formula applies even to non-proper intersections, and yields a vast generalization of Bézout theorem for arbitrary intersections of projective hypersurfaces.

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