Talk:Cone (algebraic geometry)

Todo:

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

\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^3y, x^2y^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)