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Let $X$ be the cubic surface $x_0^3=x_1x_2x_3$ in $\mathbb{P}^3$. How to find a non-zero regular differential 2-form on $X$?

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    Dear Alan, What do you mean by a *regular* $2$-form on a singular surface? Regards,2012-05-04
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    Let $X$ be an irreducible variety of dimension $n$, and let $r\in\mathbb{N}$ with $r\leq n$. A differential $r$-form $\omega$ on $X$ is regular at a point $x\in X$ if there exists an open neighbourhood $U$ of $x$ in $X$, together with functions $f_{i_1,...,i_r},g_{i_1},...,g_{i_r}\in k[U]$ $(1\leq i_1<..., such that $\omega=\sum_{1\leq i_1<... on $U$. A differential $r$-form is regular on $X$ if it is regular at every point of $X$.2012-05-05
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    Dear Alan, In what space are you computing $dg_i$? Are you just working in the sheaf of Kahler differentials? Regards,2012-05-05
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    I don't really know sheaves that well. My definition of $\mathrm{d}g$ is $(\mathrm{d}g)(x)=\mathrm{d}_xg\in\Theta_{X,x}^*$ (the cotangent space of $X$ at $x$). When $X\in\mathbb{A}^n$ and $x=(x_1,...,x_n)$, then $d_xg(t_1,...,t_n)=\sum_{i=1}^{n}\frac{\partial g}{\partial x_i}(t_i-x_i)$. This definition is from Shafarevich's book "Basic Algebraic Geometry Volume 1" (before he introduces schemes and sheaves).2012-05-05
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    Dear Alan, But what does the cotangent space mean if $X$ is singular at $x$? (This is the point of my queries.) If $X$ is smooth, then there is no ambiguity about what differentials mean, but if $X$ is singular, so that the Zariski (co)tangent space jumps in dimension at singular points, then the notion of differential is more problematic. One natural notion to use, which seems to be what you are (implictly) using is Kahler differentials. (These are objects which specialize to lie in the Zariski cotangent space of each point of their domain.) But these are still not so well behaved ...2012-05-05
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    ... in the case of singular varieties, e.g. because they are not sections of a vector bundle (just of a coherent sheaf which is not locally free when $X$ is singular), and so taking exterior powers is not such a good operation. There are other ways to generalize differential $2$-forms to singular surfaces then via taking the $2$nd exterior power of the Kahler differentials. E.g. on a smooth surface, the bundle of $2$-forms is the canonical bundle, and this can be generalized to the singular case via the notion of *dualizing sheaf*. This very often provides a better extension of the ...2012-05-05
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    ... notion of $2$-form to a singular surface than taking the $2$nd exterior power of Kahler differentials. Perhaps you can explain why you want a non-zero regular $2$-form on your surface. Regards,2012-05-05

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