001 Practice Questions

Berkeley questions

  • What are the involutions of an elliptic curve over \({\mathbf{C}}\)? 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 \(\widehat{{\mathbf{C}}}\)? McMullen

  • Let’s talk about Riemann-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 \({\rm Pic}(k[t^2,t^3])\) and ???? \(k[t^2,t^3]\subset k[t]\). Ogus

  • Find an example of a projective curve which is not rational.

  • Is \({\mathbf{P}}^1\times{\mathbf{P}}^1\) a projective variety? Prove it.

  • Find the explicit equation of the image of \({\mathbf{P}}^1\times{\mathbf{P}}^1\) under the Segre embedding \begin{align*}\psi({\mathbf{P}}^1\times{\mathbf{P}}^1)\subseteq {\mathbf{P}}^3\end{align*}

  • If the field is \({\mathbf{C}}\), the embedding in \({\mathbf{P}}^3\) 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 \(P_X(r)\)) have to do with line bundles on \({\mathbf{P}}^1\) (namely, \({\mathcal O}(3)\))? Ogus

  • What does the constant term of \(P_X(r)\) represent?

  • Let \(X\) be the twisted cubic in \({\mathbf{A}}^3\). Is \(X\) an intersection (set-theoretically) of two surfaces in \({\mathbf{P}}^3\)? Ogus

  • Is the twisted cubic in \({\mathbf{A}}^3\) the intersection (scheme-theoretically) of two surfaces in \({\mathbf{P}}^3\)? Ogus

  • What can you say about curves \(Y\leq {\mathbf{A}}^3\) and \(Y\equiv {\mathbf{A}}^1\): are they (set,scheme)-theoretically 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 \(g,h: Z\to X\), with \(Z\) and \(X\) schemes over \(Y\), via \(f: X\to Y\), 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 \({\mathbf{P}}^n\) to \({\mathbf{P}}^m\)? 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 theorem. 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 \(H^1\) 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 \(H^1(X,I) = 0\) 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 \({\mathbf{P}}^1\) or can be embedded as a quadric in \({\mathbf{P}}^2\). Ogus

  • If the base field is a finite field, can the latter case occur? Ogus

  • Calculate \(H^0({\mathbf{P}}^1,\Omega^1)\). 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 \({\mathbf{P}}^2\) defined by the equation \(X^3 + Y^3 + Z^3 = 3CXYZ\). 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 \(\operatorname{Pic}(V) ={\mathbf{Z}}/3{\mathbf{Z}}\)? Poonen

  • Does there exist a projective variety \(V\) with \(\mathop{\rm Pic}(V) = {\mathbf{Z}}/3{\mathbf{Z}}\)? Poonen

  • Is the complement of a hypersurface in \({\mathbf{P}}^2\) affine? Poonen

  • Define the geometric genus. Poonen

  • What might be the geometric genus of a singular curve? Poonen

  • Find the arithmetic genus of \(y^3=x^2 z\). Frenkel

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