[latex]
\[
\textbf{Difficult MCQs on Algebraic Geometry with Answers}
\]
\[
\textbf{Q1: What is the dimension of the variety defined by the equation } x^2 + y^2 + z^2 = 0 \text{ in } \mathbb{C}^3 \text{ (complex 3-dimensional space)?}
\]
\[
\text{(A) } 2
\]
\[
\text{(B) } 1
\]
\[
\text{(C) } 0
\]
\[
\text{(D) } 3
\]
\[
\textbf{Answer: (C) 0}
\]
\[
\textbf{Q2: Which of the following is a property of the Hilbert Nullstellensatz?}
\]
\[
\text{(A) It provides a correspondence between points of an affine variety and ideals in the coordinate ring.}
\]
\[
\text{(B) It gives a way to construct affine varieties from ideals.}
\]
\[
\text{(C) It shows that the coordinate ring of a variety is always a domain.}
\]
\[
\text{(D) It states that every affine variety is reducible.}
\]
\[
\textbf{Answer: (A) It provides a correspondence between points of an affine variety and ideals in the coordinate ring.}
\]
\[
\textbf{Q3: Which of the following spaces is not a projective variety?}
\]
\[
\text{(A) The set of solutions to a homogeneous polynomial equation in } \mathbb{P}^2.
\]
\[
\text{(B) The set of solutions to a system of linear equations in } \mathbb{P}^n.
\]
\[
\text{(C) The set of solutions to a system of polynomial equations in } \mathbb{P}^n.
\]
\[
\text{(D) The set of solutions to a non-homogeneous polynomial equation in } \mathbb{P}^2.
\]
\[
\textbf{Answer: (D) The set of solutions to a non-homogeneous polynomial equation in } \mathbb{P}^2.
\]
\[
\textbf{Q4: What does the Bezout’s theorem state?}
\]
\[
\text{(A) The number of points of intersection of two varieties equals the degree of their product.}
\]
\[
\text{(B) The number of points of intersection of two varieties equals the product of their degrees, counted with multiplicities.}
\]
\[
\text{(C) The number of points of intersection of two varieties is always finite.}
\]
\[
\text{(D) The number of points of intersection of two varieties is infinite.}
\]
\[
\textbf{Answer: (B) The number of points of intersection of two varieties equals the product of their degrees, counted with multiplicities.}
\]
\[
\textbf{Q5: The affine variety corresponding to the ideal } (x^2 + y^2 – 1) \text{ in } \mathbb{C}^2 \text{ is:}
\]
\[
\text{(A) A circle in the complex plane.}
\]
\[
\text{(B) A hyperbola in the complex plane.}
\]
\[
\text{(C) A set of two distinct points.}
\]
\[
\text{(D) A circle in real coordinates.}
\]
\[
\textbf{Answer: (C) A set of two distinct points.}
\]
\[
\textbf{Q6: What is the genus of a smooth projective curve of degree 6 in } \mathbb{P}^2?
\]
\[
\text{(A) } 4
\]
\[
\text{(B) } 5
\]
\[
\text{(C) } 6
\]
\[
\text{(D) } 7
\]
\[
\textbf{Answer: (B) 5}
\]
\[
\textbf{Q7: Which of the following is a correct statement about the Riemann-Roch theorem?}
\]
\[
\text{(A) It provides a formula to compute the number of independent sections of a line bundle.}
\]
\[
\text{(B) It gives a way to calculate the Euler characteristic of a variety.}
\]
\[
\text{(C) It provides a way to compute the dimension of the cohomology groups.}
\]
\[
\text{(D) It states that every algebraic variety has a finite number of connected components.}
\]
\[
\textbf{Answer: (A) It provides a formula to compute the number of independent sections of a line bundle.}
\]
\[
\textbf{Q8: What is the Picard group of a variety?}
\]
\[
\text{(A) The group of all divisors on the variety.}
\]
\[
\text{(B) The group of all line bundles on the variety.}
\]
\[
\text{(C) The group of all projective varieties on the variety.}
\]
\[
\text{(D) The group of all affine varieties on the variety.}
\]
\[
\textbf{Answer: (B) The group of all line bundles on the variety.}
\]
\[
\textbf{Q9: The Hilbert scheme parametrizes:}
\]
\[
\text{(A) Families of algebraic varieties.}
\]
\[
\text{(B) Families of schemes.}
\]
\[
\text{(C) Families of modules over a ring.}
\]
\[
\text{(D) Families of coherent sheaves.}
\]
\[
\textbf{Answer: (A) Families of algebraic varieties.}
\]
\[
\textbf{Q10: What is the degree of the map from the projective plane } \mathbb{P}^2 \text{ to itself given by } (x : y : z) \mapsto (x^2 : xy : y^2)?
\]
\[
\text{(A) } 2
\]
\[
\text{(B) } 3
\]
\[
\text{(C) } 4
\]
\[
\text{(D) } 5
\]
\[
\textbf{Answer: (A) 2}
\]