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Grassmann bundle

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an <a href="Algebraic_scheme" class="mw-redirect" title="Algebraic scheme">algebraic scheme</a> X is a scheme over X:

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<annotation encoding="application/x-tex">{\displaystyle p:G_{d}(E)\to X}</annotation>

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</math><img src="./0b684c9570d26d285f1e9d614b55ee484c8af070.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:15.294ex; height:2.843ex;" alt="{\displaystyle p:G_{d}(E)\to X}" loading="lazy"></dd></dl>

such that the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-1}(x)=G_{d}(E_{x})}">

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<annotation encoding="application/x-tex">{\displaystyle p^{-1}(x)=G_{d}(E_{x})}</annotation>

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</math><img src="./ac6810dc437e8946063256f624b911bb1360d9ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:17.445ex; height:3.176ex;" alt="{\displaystyle p^{-1}(x)=G_{d}(E_{x})}" loading="lazy"> is the <a href="Grassmannian" title="Grassmannian">Grassmannian</a> of the d-dimensional vector subspaces of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{x}}">

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<annotation encoding="application/x-tex">{\displaystyle E_{x}}</annotation>

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</math><img src="./029e49fbec18ece71cdd1e68bc478444e2c99d30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.888ex; height:2.509ex;" alt="{\displaystyle E_{x}}" loading="lazy">. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{1}(E)=\mathbb {P} (E)}">

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<annotation encoding="application/x-tex">{\displaystyle G_{1}(E)=\mathbb {P} (E)}</annotation>

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</math><img src="./f66ec867482dbbc771c85be4b0d78faab9ed7bee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.57ex; height:2.843ex;" alt="{\displaystyle G_{1}(E)=\mathbb {P} (E)}" loading="lazy"> is the <a href="Projective_bundle" title="Projective bundle">projective bundle</a> of E. In the other direction, a Grassmann bundle is a special case of a (partial) <a href="Flag_bundle" title="Flag bundle">flag bundle</a>. Concretely, the Grassmann bundle can be constructed as a <a href="Quot_scheme" title="Quot scheme">Quot scheme</a>.

Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or <a href="Tautological_subbundle" class="mw-redirect" title="Tautological subbundle">tautological subbundle</a> S and universal quotient bundle Q that fit into

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<annotation encoding="application/x-tex">{\displaystyle 0\to S\to p^{*}E\to Q\to 0}</annotation>

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</math><img src="./2beb39c46c2a411ddc0a1b65874cdc568d1931ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.118ex; height:2.676ex;" alt="{\displaystyle 0\to S\to p^{*}E\to Q\to 0}" loading="lazy">.</dd></dl>

Specifically, if V is in the fiber p−1(x), then the fiber of S over V is V itself; thus, S has rank r = d = dim(V) and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge ^{d}S}">

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<annotation encoding="application/x-tex">{\displaystyle \wedge ^{d}S}</annotation>

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</math><img src="./af49a5f3ed3bac640fb610fabfa35800eff7dd80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.142ex; height:2.676ex;" alt="{\displaystyle \wedge ^{d}S}" loading="lazy"> is the <a href="Determinant_line_bundle" class="mw-redirect" title="Determinant line bundle">determinant line bundle</a>. Now, by the universal property of a projective bundle, the injection <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \wedge ^{r}S\to p^{*}(\wedge ^{r}E)}">

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<annotation encoding="application/x-tex">{\displaystyle \wedge ^{r}S\to p^{*}(\wedge ^{r}E)}</annotation>

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</math><img src="./b2814384a57c0be1d9a547eb4ddfb429df085a3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.97ex; height:2.843ex;" alt="{\displaystyle \wedge ^{r}S\to p^{*}(\wedge ^{r}E)}" loading="lazy"> corresponds to the morphism over X:

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<annotation encoding="application/x-tex">{\displaystyle G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)}</annotation>

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</math><img src="./c506a28bda1534be8a6594856256b4d991c46d8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.647ex; height:2.843ex;" alt="{\displaystyle G_{d}(E)\to \mathbb {P} (\wedge ^{r}E)}" loading="lazy">,</dd></dl>

which is nothing but a family of <a href="Pl%C3%BCcker_embedding" title="Plücker embedding">Plücker embeddings</a>.

The <a href="Relative_tangent_bundle" class="mw-redirect" title="Relative tangent bundle">relative tangent bundle</a> TGd(E)/X of Gd(E) is given by<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle T_{G_{d}(E)/X}=\operatorname {Hom} (S,Q)=S^{\vee }\otimes Q,}</annotation>

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</math><img src="./6fa915daaa72f488dc6934b0c78b43cf8e3182c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:33.896ex; height:3.343ex;" alt="{\displaystyle T_{G_{d}(E)/X}=\operatorname {Hom} (S,Q)=S^{\vee }\otimes Q,}" loading="lazy"></dd></dl>

which morally is given by the <a href="Second_fundamental_form" title="Second fundamental form">second fundamental form</a>. In the case d = 1, it is given as follows: if V is a finite-dimensional vector space, then for each line <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l}">

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<annotation encoding="application/x-tex">{\displaystyle l}</annotation>

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</math><img src="./829091f745070b9eb97a80244129025440a1cfac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.693ex; height:2.176ex;" alt="{\displaystyle l}" loading="lazy"> in V passing through the origin (a point of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {P} (V)}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {P} (V)}</annotation>

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</math><img src="./8afe30e988d193afbadbc380123eb6cc6d9007dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.017ex; height:2.843ex;" alt="{\displaystyle \mathbb {P} (V)}" loading="lazy">), there is the natural identification (see <a href="Chern_class#Complex_projective_space" title="Chern class">Chern class#Complex projective space</a> for example):

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (l,V/l)=T_{l}\mathbb {P} (V)}</annotation>

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</math><img src="./98a8341145bbf2dabd29bdc7503cc36250674f96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.216ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (l,V/l)=T_{l}\mathbb {P} (V)}" loading="lazy"></dd></dl>

and the above is the family-version of this identification. (The general care is a generalization of this.)

In the case d = 1, the early exact sequence tensored with the dual of S = O(-1) gives:

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<annotation encoding="application/x-tex">{\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} (E)}\to p^{*}E\otimes {\mathcal {O}}_{\mathbb {P} (E)}(1)\to T_{\mathbb {P} (E)/X}\to 0}</annotation>

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</math><img src="./e4b24c6441eb59d5b0865075764f5719095a2139.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:45.187ex; height:3.176ex;" alt="{\displaystyle 0\to {\mathcal {O}}_{\mathbb {P} (E)}\to p^{*}E\otimes {\mathcal {O}}_{\mathbb {P} (E)}(1)\to T_{\mathbb {P} (E)/X}\to 0}" loading="lazy">,</dd></dl>

which is the relative version of the <a href="Euler_sequence" title="Euler sequence">Euler sequence</a>.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> <a href="#CITEREFFulton1998">Fulton 1998</a>, Appendix B.5.8

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</style><cite id="CITEREFEisenbudJoe2016" class="citation cs2">Eisenbud, David; Joe, Harris (2016), 3264 and All That: A Second Course in Algebraic Geometry, C. U.P., <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1107602724</bdi></cite></li>

<li><cite id="CITEREFFulton1998" class="citation cs2">Fulton, William (1998), Intersection theory, <a href="Ergebnisse_der_Mathematik_und_ihrer_Grenzgebiete" title="Ergebnisse der Mathematik und ihrer Grenzgebiete">Ergebnisse der Mathematik und ihrer Grenzgebiete</a>. 3. Folge., vol. 2 (2nd ed.), Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-62046-4</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1644323">1644323</a></cite></li></ul></div><div><div class="zim-footer">

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