- - - - # Questions - [ ] Are differentials quasicoherent? **Ogus** - [ ] Calculate $\operatorname{Pic}\left(k\left[t^{2}, t^{3}\right]\right) . k\left[t^{2}, t^{3}\right] \subset k[t]$. **Ogus** - [ ] Calculate $H^0(\PP^1,\Omega^1)$. **Poonen** - [ ] Compute the dimension of the space of holomorphic differentials on a Riemann surface of genus $g$. **Wodzicki** - [ ] Define differentials. **Ogus** - [ ] Define separated morphism. **Ogus** - [ ] Define the geometric genus. **Poonen** - [ ] Describe Weil divisors and Cartier divisors on curves. **Ogus** - [ ] Does the genus of a curve depend on the embedding? **Hartshorne** - [ ] Does there exist a *projective* variety $V$ with $\mathop{\rm Pic}(V) = \ZZ/3\ZZ$? **Poonen** - [ ] Does there exist a variety $V$ with $\Pic(V) =\ZZ/3\ZZ$? **Poonen** - [ ] Do you know what quasi-separated means? **Ogus** - [ ] Find an example of a projective curve which is not rational. - [ ] Find the arithmetic genus of $y^3=x^2 z$. **Frenkel** - [ ] Find the explicit equation of the image of $\mathbb{P}^{1} \times \mathbb{P}^{1}$ under the Segre embedding $$\psi\left(\mathbb{P}^{1} \times \mathbb{P}^{1}\right) \subseteq \mathbb{P}^{3}$$ - [ ] If the field is $\CC$, the embedding in $\PP^3$ is the $4$-dimensional manifold. Compute the intersection form. - [ ] Find the singularities - if any - of the curve in $\mathbb{P}^{2}$ defined by the equation $X^{3}+Y^{3}+$ $Z^{3}=3 C X Y Z$. **Ogus** - [ ] Give an example of a non-separated morphism. **Poonen** - [ ] Give two criteria for a curve to be nonsingular (over an algebraically closed field). **Ogus** - [ ] How can you tell if a scheme is affine? **Ogus** - [ ] Can you weaken the Noetherian hypothesis in Serre's criterion for affineness? **Ogus** - [ ] How do you get a Weil divisor from an element $f \in K^*$, in the canonical isomorphism? **Ogus** - [ ] How do you use Hurwitz's formula to calculate the genus of a given curve? **Coleman** - [ ] 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? - [ ] If the base field is a finite field, can the latter case occur? **Ogus** - [ ] Is $\mathbb{P}^{1} \times \mathbb{P}^{1}$ a projective variety? Prove it. - [ ] Is the complement of a hypersurface in $\PP^2$ affine? **Poonen** - [ ] Let $X$ be the twisted cubic in $\AA^3$. Is $X$ an intersection (set-theoretically) of two surfaces in $\PP^3$? **Ogus** - [ ] 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** - [ ] Name a good property of separated morphisms. **Ogus** - [ ] What would be the analogue for quasiseparated in place of separated? **Ogus** - [ ] Now can you prove the weak version of Riemann-Hurwitz? **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** - [ ] Show that a hypersurface defined by an equation of degree $d$ has degree $d$. **Sturmfels** - [ ] State Abel's theorem. **Wodzicki** - [ ] State Riemann-Roch. **Wodzicki** - [ ] What are the involutions of an elliptic curve over $\mathbb{C}$ ? **McMullen** - [ ] What are the fixed points of this involution? **McMullen** - [ ] What quotient arises from this involution? **McMullen - [ ] So how can you show this quotient is $\hat{\mathbb{C}}$ ? **McMullen** - [ ] What can you say about curves $Y\leq \AA^3$ and $Y\equiv \AA^1$: are they (set,scheme)-theoretically intersections of two surfaces? **Ogus, later recanted** - [ ] What can you say about curves of genus $0$? **Ogus** - [ ] Prove that such a curve is always isomorphic to $\PP^1$ or can be embedded as a quadric in $\PP^2$. **Ogus** - [ ] What can you say about curves over perfect fields? **Coleman** - [ ] 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** - [ ] What can you say about the dimension of the image of a map from $\mathbb{P}^{n}$ to $\mathbb{P}^{m}$ ? - [ ] What does the constant term of $P_{X}(r)$ represent? - [ ] What does the degree (leading term of $P_{X}(r)$ ) have to do with line bundles on $\mathbb{P}^{1}$ (namely, $\mathcal{O}(3)$ )? **Ogus** - [ ] What does the going up theorem mean in algebraic geometry? **Hartshorne** - [ ] What is a normal domain? How is this related to regular local rings? **Ogus** - [ ] What is a scheme? **Ogus** - [ ] What is the connection between $H^{1}$ and line bundles? - [ ] What is the degree of a divisor? **Ogus** - [ ] What is the genus of a curve? **Hartshorne** - [ ] 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 might be the geometric genus of a singular curve? **Poonen** - [ ] When is a canonical divisor very ample? **Wodzicki**