Algebraic Geometry MCQs

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