Algebraic Geometry

From MGSA
Jump to: navigation, search
  • What are the involutions of an elliptic curve over {\mathbb C}? (McMullen)
  • What quotient arises from this involution? (McMullen)
  • What are the fixed points of this involution? (McMullen)
  • So how can you show this quotient is \hat{{\mathbb C}}? (McMullen)
  • Let's talk about Riemman-Hurwitz. Given a nonconstant map between curves over k, is there an associated map on differentials? A resulting exact sequence? (Ogus)
  • Is the exact sequence short exact in this case? (Ogus)
  • Now can you prove the weak version of Riemann-Hurwitz? (Ogus)
  • Calculate Pic(k[t2,t3]). k[t^2,t^3]\subset k[t]. (Ogus)
  • Find an example of a projective curve which is not rational.
  • Is {\mathbb P}^1\times{\mathbb P}^1 a projective variety? Prove it.
  • Find the explicit equation of the image of {\mathbb P}^1\times{\mathbb P}^1 under the Segre embedding \psi({\mathbb P}^1\times{\mathbb P}^1)\subseteq {\mathbb P}^3.
  • If the field is {\mathbb C}, the embedding in {\mathbb P}^3 is the 4-dimensional manifold. Compute the intersection form.
  • How do you use Hurwitz's formula to calculate the genus of a given curve? (Coleman)
  • What can you say about curves over perfect fields? (Coleman)
  • Show that a hypersurface defined by an equation of degree d has degree d. (Sturmfels)
  • What does the degree (leading term of PX(r)) have to do with line bundles on {\mathbb P}^1 (namely, {\mathcal O}(3))? (Ogus)
  • What does the constant term of PX(r) represent?
  • Let X be the twisted cubic in {\mathbb A}^3. Is X an intersection (set-theoretically) of two surfaces in {\mathbb P}^3? (Ogus)
  • Is the twisted cubic in {\mathbb A}^3 the intersection (scheme-theoretically) of two surfaces in {\mathbb P}^3? (Ogus)
  • What can you say about curves Y\leq {\mathbb A}^3 and Y\equiv {\mathbb A}^1: are they (set,scheme)-theoreticallly intersections of two surfaces? (Ogus, later recanted)
  • Define separated morphism. (Ogus)
  • Give an example of a non-separated morphism. (Poonen)
  • Do you know what quasi-separated means? (Ogus)
  • Name a good property of separated morphisms. (Ogus)
    • What would be the analogue for quasiseparated in place of separated? (Ogus)
  • What can you say about separated schemes? (Ogus)
    • Say g,h\colon Z\to X, with Z and X schemes over Y, via f\colon X\to Y, and g and h agreeing on an open dense subset of Z. What can be said if f is separated? If Z is reduced? (Ogus)
    • Give examples where Z is not reduced or f not separated and g\not= h. (Ogus)
  • Define differentials. (Ogus)
  • Are differentials quasicoherent? (Ogus)
  • What does the going up theorem mean in algebraic geometry? (Hartshorne)
  • What can you say about the dimension of the image of a map from {\mathbb P}^n to {\mathbb P}^m? (Hartshorne)
  • What is the genus of a curve? (Hartshorne)
  • Does the genus of a curve depend on the embedding? (Hartshorne)
  • When is a canonical divisor very ample? (Wodzicki)
  • State Riemann-Roch. (Wodzicki)
  • Compute the dimension of the space of holomorphic differentials on a Riemann surface of genus g. (Wodzicki)
  • State Abel's theorm. (Wodzicki)
  • What is the significance of the Jacobian? What kind of map is the Abel-Jacobi map? What is it in the case of genus 1? (Wodzicki)
  • What is the connection between H1 and line bundles? (Wodzicki)
  • What is a scheme? (Ogus)
  • How can you tell if a scheme is affine? (Ogus)
  • Can you weaken the Noetherian hypothesis in Serre's criterion for affineness? (Ogus)
  • Prove that if X is a Noetherian scheme such that H1(X,I) = 0 for all coherent sheaves of ideals I then X is affine. (Ogus)
  • Can you give an example where the theorem is false if we drop the quasi-compactness assumption? (Ogus)
  • What can you say about curves of genus 0? (Ogus)
  • Prove that such a curve is always isomorphic to {\mathbb P}^1 or can be embedded as a quadric in {\mathbb P}^2. (Ogus)
  • If the base field is a finite field, can the latter case occur? (Ogus)
  • Calculate H^0({\mathbb P}^1,\Omega^1). (Poonen)
  • If f(x,y) and g(x,y) are two polynomials such that the curves they define have infinitely many points in common, is it true that they have a common factor?
  • Give two criteria for a curve to be nonsingular (over an algebraically closed field). (Ogus)
  • What is a normal domain? How is this related to regular local rings? (Ogus)
  • Find the singularities—if any—of the curve in {\mathbb P}^2 defined by the equation X3 + Y3 + Z3 = 3CXYZ. (Ogus)
  • Describe Weil divisors and Cartier divisors on curves. (Ogus)
  • How do you get a Weil divisor from an element f \in K^*, in the canonical isomorphism? (Ogus)
  • What is the degree of a divisor? (Ogus)
  • Does there exist a variety V with \mathop{\rm Pic}(V) = \Z/3\Z? (Poonen)
  • Does there exist a projective variety V with \mathop{\rm Pic}(V) = \Z/3\Z? (Poonen)
  • Is the complement of a hypersurface in {\mathbb P}^2 affine? (Poonen)
  • Define the geometric genus. (Poonen)
  • What might be the geometric genus of a singular curve? (Poonen)
  • Find the arithmetic genus of y3 = x2z. (Frenkel)
  • Define sheaf cohomology. What's a right derived functor? (Olsson)
  • Let E be the curve in {\mathbb P}^2 defined by y2 = x3 − 1. Compute the cohomology of the structure sheaf \mathcal O_E. (Olsson)
  • Define projective morphisms and what are they good for? What's a morphism that isn't projective? (Eisenbud)
  • Define Cartier and Weil divisors and relate them to each other. Do you know a Weil divisor which isn't Cartier? Compute the Picard group and class group of the cone over a conic. (Eisenbud)
  • What can you say about curves of degree 4 in P^3? What if they are contained in a plane? What if they are singular? (Eisenbud)
  • Let X be a quartic surface in P^3. Does X contain a curve with negative self intersection. i.e. can the normal bundle to the curve have negative degree? (Eisenbud)

This page was originally derived from this TeX file.

Retrieved from "http://www.ocf.berkeley.edu/~mgsa/wiki/Algebraic_Geometry"
Personal tools