Algebraic geometry

Which of these subsets of the complex line $\mathbb{A}^1 = \mathbb{C}$ are closed in the Zariski topology? (Choose all that apply.)

A smooth cubic curve and a conic in the complex projective plane have no common component. In how many points do they meet, counted with multiplicity?

Over $\mathbb{C}$, let $X = V(x^2 y,\, x y^2) \subseteq \mathbb{A}^2$. Which ideal is $I(X)$?

Which argument correctly proves that the hyperbola $H = V(xy - 1)$ is not isomorphic to the affine line $\mathbb{A}^1$ (over $\mathbb{C}$)?

How many points does the projective plane $\mathbb{P}^2(\mathbb{F}_5)$ have?

The concentric circles $x^2 + y^2 = 1$ and $x^2 + y^2 = 4$ have no common point in $\mathbb{C}^2$. Bézout's theorem promises $4$ intersection points in $\mathbb{P}^2(\mathbb{C})$. Where are they?

The morphism $\varphi : \mathbb{A}^1 \to C = V(y^2 - x^3)$, $t \mapsto (t^2, t^3)$, is a bijection. Which statement explains why it is not an isomorphism?

Which Lean statement formalises Proposition 1's identity "the zero set of a product of ideals is the union of the zero sets"? (zeroLocus K I is the set of points of $K^\sigma$ where every polynomial in I vanishes.)

Write projective_count(f, p) returning the number of points of $\mathbb{P}^2(\mathbb{F}_p)$ at which the homogeneous polynomial f(x, y, z) (a Python function with integer values) vanishes modulo $p$. Use one representative per point: $(x, y, 1)$, $(x, 1, 0)$ and $(1, 0, 0)$.