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I would mention B\'ezout's Bézout's theorem. Forgetting the complicated general definition of the intersection index one of the consequences is: whenever two curves in $\mathbf{P}^2(K), K$ an algebraically closed field have no common component, the number of the intersection points is finite and is always at most the product of the degrees. Moreover, it is the product of the degrees provided no intersection point is a singular point and all intersections are transversal. To state this over $\mathbf{C}$ we essentially only need multivariable calculus. But a proof requires a bit of algebraic geometry.

Let me also mention two algebraic geometry books that present precisely the kind of material people in other areas are likely to use. One is "Algebraic geometry, a first course" by Joe Harris; the other is "Undergraduate algebraic geometry" by Miles Reid. If memory serves, it says somewhere in the latter book that it covers (together with Atiyah-MacDonald) all algebraic geometry questions that the author was ever asked by his colleagues who specialize in other areas.
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I would mention B\'ezout's theorem. Forgetting the complicated general definition of the intersection index one of the consequences is: whenever two curves in $\mathbf{P}^2(K), K$ an algebraically closed field have no common component, the number of the intersection points is finite and is always at most the product of the degrees. Moreover, it is the product of the degrees provided no intersection point is a singular point and all intersections are transversal. To state this over $\mathbf{C}$ we essentially only need multivariable calculus. But a proof requires a bit of algebraic geometry.

Let me also mention two algebraic geometry books that present precisely the kind of material people in other areas are likely to use. One is "Algebraic geometry, a first course" by Joe Harris; the other is "Undergraduate algebraic geometry" by Miles Reid. If memory serves, it says somewhere in the latter book that it covers (together with Atiyah-MacDonald) all algebraic geometry questions that the author was ever asked by his colleagues who specialize in other areas.