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What's the definition of canonical divisor(or whatever related concept) on singular curve with nodal point. More generally, what the definition of canonical divisor on a singular variety X, which is the union of 2 smooth varieties intersecting with each other

xin fu
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  • have you consulted Serre's Groupes Algebriques et corps de classes? briefly, since the curve is a limit of a smooth curve as a homology cycle shrinks to a point, a canonical divisor is the divisor of a form which is such a limit. Consider what is the limit of a holomorphic form with a period around the vanishing cycle. Hint: it becomes meromorphic on the normalization, but with period around the marked pointover the node. – roy smith Nov 26 '19 at 03:03
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    What's the/your definition of a divisor on such a variety? (cf. https://mathoverflow.net/a/46663/10076) – Sándor Kovács Nov 26 '19 at 19:38
  • Nodes are Gorenstein singularities, in particular the dualizing sheaf is a line bundle. If $X$ is a nodal curve with smooth components, then its canonical bundle restricted on each component $X_i$ is $\omega_{X_i}(P_1 + \dots + P_r)$, where $P_1, \dots, P_r$ are intersection points with other components. As an exercise one shows that if $X$ is a nodal degree $3$ curve in $P^2$, then its canonical bundle is trivial. See discussion here for the dualizing sheaf of nodal curves and stability: http://people.math.umass.edu/~tevelev/797_2017/arie_seba.pdf – Evgeny Shinder Nov 26 '19 at 21:49
  • A similar discussion applies in higher dimension for smooth varieties intersecting along smooth divisors. However, if components intersect in higher codimension, then it becomes much more complicated. For instance, union of two planes intersecting in a point is famously non Cohen-Macaulay, and hence non Gorenstein. – Evgeny Shinder Nov 26 '19 at 21:52
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    Just for the record: It is not enough if smooth components intersect along a smooth divisor to have Gorenstein singularities. For instance, the union of three coordinate axis in $3$-dimensional space has smooth irreducible components intersecting along a subvariety which is a smooth divisor in each component, but it is not Gorenstein. – Sándor Kovács Nov 27 '19 at 05:12
  • Of course, a simple normal crossing variety is indeed Gorenstein and hence has a canonical sheaf which is a line bundle. However, and this was my point in my previous comment, if the space is not normal, then it is dangerous to talk about divisors. One can talk about Weil divisorial sheaves or in other words reflexive sheaves of rank $1$, but this is not the same as a divisor. These sheaves (as the canonical sheaf of a Gorenstein scheme) correspond to linear equivalence classes of divisors not to divisors. (See the link in my previous comment for more on this). :) – Sándor Kovács Nov 27 '19 at 05:15
  • @SándorKovács, yes, thank you, I meant simple normal crossing intersections, in this case two smooth components intersecting transversely along a smooth divisor. I looked up your explanation about divisors, which I found very helpful. I guess my point was that canonical sheaf should be used instead of a canonical divisor if the variety is reducible. – Evgeny Shinder Nov 28 '19 at 14:57
  • @SándorKovács@EvgenyShinder Great thanks!! – xin fu Nov 28 '19 at 19:52

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