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Consider the Hilbert scheme of curves in $P^3$ with genus $g$ and degree $d$, $H_{g,d}$. Is this rational for some $g$ and $d$? Edit: For which $(g,d)$ is this rational?

By an admissable subcategory $A$ in a triangulated category $B$, I mean a triangulated subcategory that has $A \oplus B \in A$, then $A$, $B \in A$, and so that there is either a right or left adjoint …

Can someone give me an indication or reference for a situation in which the geometry of some moduli stack is "used"? I know that intersection theory and cohomologies can be developed on algebraic stac …

It is a well known fact that a smooth cubic surface in $P^3$ contains 27 lines. One proof proceeds by moving through the parameter space $U$ of smooth cubics until one reaches an cubic that can be und …

I heard a reference to a statement like: Suppose $A$ is an (Abelian?) category of homological dimension one, then the stack of objects of $A$ is smooth. (I am not really sure what the stack of object …

Ehresmann's theorem for manifolds states: If $f : X \to Y$ is a proper submersion, then $X$ is a locally trivial fibration on $Y$. Some sources I am reading (Lazarsfeld Positivity in Algebraic Geomet …

Background: Let $P$ be a integral polytope, and $X_P$ the toric variety associated to the normal fan. $X_P$ is always projective, because the collection of characters corresponding to the points $\ma …

I think I found an a example of a family of toric varieties whose minimal embedding dimension is exponential in their dimension. The idea is to produce a singular point with a high dimensional tangent …

I was describing to a friend the result that a compact Hausdorff space is determined up to homeomorphism up to by its ring of continuous functions, and he asked how one could see the genus of a surfac …