Talk:Cone (algebraic geometry)

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

$\Gamma ({\mathcal {O}}(4))=\mathbb {C} \cdot \langle x^{4},x^{3}y,x^{2}y^{2},xy^{3},y^{4},z\rangle$

Link broken

The link for Fantechi's lecture notes seems broken, does anyone find them anywhere else? 2A02:8071:B69E:9300:5471:1A9E:8287:91F8 (talk) 10:34, 1 February 2019 (UTC)