Talk:Cone (algebraic geometry)

The terminology projective cone seems to be somewhat misleading and it is not clear how standard it is. It is used in Fulton, but other sources (e.g. Ravi Vakil's notes) seem to use the term to refer to the projective completion. It might be more appropriate to use the term projectivized cone, or projectivization of the cone. NickTK (talk) 16:28, 13 October 2024 (UTC)[reply]

Todo:

give motivation for cone construction and applications (such as with virtual fundamental classes)
compute explicit examples (maybe mention macaulay2 functions...)

Fantechi gives a few good examples in her notes and differentiates between abelian and regular cones. One can generalize her computations though: given a complete intersection ideal $(g_{1},g_{2})$ of ${\mathcal {O}}_{M}$ we can consider the cone of the ideal $(f)(g_{1},g_{2})$ . If we write the quotient ${\mathcal {O}}_{X}={\mathcal {O}}_{M}/{\mathcal {I}}$ , then the associated cone is given by the relative spec of the sheaf of algebras

{\frac {{\mathcal {O}}_{X}[a,b]}{(ag_{2}-bg_{1})}}

This example should motivate the examples for sheaves on DM-stacks. The main geometric examples I know of are the weighted projective stacks $\mathbb {P} (a_{1},\ldots ,a_{k})$ with $gcd(a_{i},a_{j})=1$ for $i\neq j$ and complete intersection substacks. If we take the ring $\mathbb {C} [x_{1},\ldots ,x_{k}]$ then the weights determine a grading of the ring. If we take the stacky proj, then we get a DM-stack. It should be obvious from here that there are associated sheaves from the $\mathbb {G} _{m}$ action, and they are what you would expect. For example, on $\mathbb {P} (1,1,4)$ we have