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I know the definition of the ring of differential operators, by using the smooth coverings. But I do not know how to calculate the global section. Can anyone help explain that why the global section …

A naive way to see why we need twist by square root of $\omega_{S}$ is the following. Given $L:\omega_{S}^{1/2}\rightarrow \omega_{S}^{1/2}$, we can form its adjoint $L^{t}:\omega_{S}^{1/2}\rightarrow …

I am reading De Cataldo and Migliorini's beautiful paper "the hard Lefschetz theorem and the topology of semismall maps", where they construct an intersection form for each relevant strata, and whose …

Suppose $X$ is a projective smooth, geometrically connected, hyperelliptic curves over a field k, we may ask $X(k)\neq\varnothing$. I want to know how to compute the $H^{0}(X,\Omega_{X}^{1})$. And th …

Let $\mathcal{O}$ be a discrete valuation ring, and $\mathcal{G}$ is a flat group scheme over $\mathcal{O}$, we may assume the generic fiber is reductive. Then can we define its Lie algebra, and wheth …

I think when you mention a smooth map, the condition seems appear in the category of diffential geometry, which is quite different. The concept of 1 dimensional separated scheme (since we consider cur …

In Chriss and Ginzburg's fantastic book 'representation theory and complex geometry', they use the following Thom Isomorphism: $\pi:E\rightarrow X$, is a G-equivariant affine linear bundle, then $\pi …