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wdecker
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topic: algebraic geometry
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Looks good to me, thanks! I've added a few comments. |
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pbelmans
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Apr 21, 2026
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Looks good to me, thanks! I've added a few comments.
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| (or, equivalently, its total Chern class). Abstract bundles support the standard operations on vector bundles | ||
| in algebraic geometry: direct sum, tensor product, duals, determinant bundles, exterior and symmetric powers, | ||
| as well as pullback and pushforward along abstract variety maps. | ||
| An *abstract bundle* on an abstract variety $X$ is determined its Chern character (or, equivalently, by its rank and total Chern class). |
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is determined by its Chern character
| In Oscar, abstract flag bundles are constructed using the function `flag_bundle` which, in particular, | ||
| allows one to implement abstract projective spaces, Grassmannians, flag varieties, and projective bundles. | ||
| In addition, there are specialized constructors for the latter varieties which may or may not rely on | ||
| In addition, there are specialized constructors for the latter varieties some of which which rely on |
| The authors of `Schubert2` are Daniel R. Grayson, Michael E. Stillman, Stein A. Strømme, David Eisenbud, and Charley Crissman, | ||
| while `Chow` is due to Manfred Lehn and Christoph Sorger. `Schubert3` as well as the `Singular` library `schubert.lib` is due | ||
| to Dang Tuan Hiep. All this work, including ours, is inspired by the `Maple` package `Schubert` written | ||
| to Dang Tuan Hiep. All this work, including ours, is inspired by the afore mentioned `Maple` package `Schubert` written |
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| If `X` has been given a polarization $\mathcal O_X(1)$, return the line bundle $\mathcal O_X(n)$ on `X`. | ||
| If `X` has been given a polarization representing the first Chern class of an ample | ||
| line bundle $\mathcal O_X(1)$ on `X`, return the line bundle $\mathcal O_X(n) := O_X(1)^{\otimes n}$. |
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in the definition, the second O_X needs a \mathcal prefix.
| a ring homomorphism that identifies the Grothendieck ring (after tensoring with $\mathbb Q$) | ||
| with the Chow ring. | ||
| a ring homomorphism that maps the Grothendieck ring (after tensoring with $\mathbb Q$) | ||
| onto the Chow ring. |
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this should maybe explicitly say numerical Chow ring, because for rational equivalence it is an isomorphism, so the reader who is skimming things might get triggered by a burst of confusion
| Then, since $X$ is supposed to be smooth, the natural map $\mathrm{K}^0(X)\rightarrow\mathrm{K}_0(X)$ is an isomorphism | ||
| (every coherent sheaf on $X$ has a finite resolution by locally free sheaves). We may, thus, speak of coherent sheaves | ||
| given by *virtual vector bundles*, that is, by elements of $\mathrm{K}^0(X)$. Hence, we can use the concept of abstract | ||
| given by *virtual vector bundles*, that is, by elements of $\mathrm{K}^0(X)_{\mathbb Q}$. Hence, we can use the concept of abstract |
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why do we need \mathbb{Q}-coefficients here, when talking about virtual vector bundles?
pbelmans
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Apr 21, 2026
| bundles. The Chern character $\mathrm{ch}\colon\mathrm{K}^0(X) \to \mathrm{N}^*(X)_{\mathbb Q}$ is | ||
| a ring homomorphism that maps the Grothendieck ring (after tensoring with $\mathbb Q$) | ||
| onto the Chow ring. | ||
| bundles. After tensoring with $\mathbb Q$, by Grothendieck--Riemann--Roch, the |
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Major pedantry, but there's a double space here.
pbelmans
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Apr 21, 2026
| for an example where the top-dimensional part of the constructed ring is more than 1-dimensional. | ||
| !!! warning | ||
| Recall from the introduction to this chapter that in many cases, there is no algorithm for | ||
| computing all generators of the Chow ring (see [Example: Cubic surfaces](@ref)). In addition note |
jankoboehm
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Apr 22, 2026
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@pbelmans This covers most of what we discussed in the chat and a bit more. One much smaller PR will follow. Then I would be happy to move things out of experimental.