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29f0111d84147e58240778a1ea211be379d7f337 | subsection | 14 | 144 | The moduli space of matroids | Then by the definition of the pullback of a matroid bundle, we have (\circ )^\ast ({\mathcal {M}}_\textup {univ})=^\ast (^\ast ({\mathcal {M}}_\textup {univ})), which establishes the functoriality of the bijection \Phi . This completes the proof of the theorem. | {
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4d066930b84ac2745266650d95fd5d7eccf1b0f6 | subsection | 15 | 144 | Duality | One of the fundamental features of matroid theory is that every matroid (with coefficients) comes with a canonical dual matroid. This extends to matroid bundles, and, in fact, the duality is derived from a duality between the moduli spaces.Theorem 1.4
Let E be a non-empty finite ordered set, r\leqslant \#E a natural n... | {
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5a52ebb4f51b4f50ae8afafd4ed442433e958326 | subsection | 16 | 144 | Duality | Then the Plücker relation given by an (r-1)-subset I and an (r+1)-subset J of E is0 \quad \leqslant \quad \sum _{i\in J-I} \ ^{(i,I)+(i,J)} \ \cdot \ x_{I\cup \lbrace i\rbrace } \ \cdot \ x_{J-\lbrace i\rbrace }.Note that(I\cup \lbrace i\rbrace )^c=I^c-\lbrace i\rbrace , \quad (J-\lbrace i\rbrace )^c=J^c\cup \lbrace i\... | {
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2e0bbf8530e7ba8c40445eb3edc8bb98e38a69ec | subsection | 17 | 144 | Duality | The dual of \Delta with respect to is the function\textstyle \begin{array}{cccc}
\Delta _^{\mbox{[-8]$\vee $}}: & \binom{E}{r^{\mbox{[-8]$\vee $}}} & \longrightarrow & \Gamma (X,{\mathcal {L}}) \\
& I & \longmapsto & _{\mathcal {L}}^\#\circ \Delta (I^c)
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b5df5e64456bce28f9b795535a407c71de0af7f1 | subsection | 18 | 144 | Duality | Then the dual \Delta _^{\mbox{[-8]$\vee $}} of \Delta with respect to is a Grassmann-Plücker function and {\mathcal {M}}_^{\mbox{[-8]$\vee $}} is the matroid bundle on X whose characteristic morphism is_{{\mathcal {M}}_^{\mbox{[-8]$\vee $}}} \ = \ ^{\mbox{[-8]$\vee $}}\circ _{\mathcal {M}}\circ : \ X \ \stackrel{}{\lon... | {
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658f2765bc6cf313c72b4f9b4dd5c99f66bf2925 | subsection | 19 | 144 | Duality | The partial order \leqslant is the smallest additive and multiplicative partial order that contains the partial order of \Gamma (X,{\mathcal {L}}^{\otimes i})^+ for every i\geqslant 0.Since \Delta is a Grassmann-Plücker function, the association x_I\mapsto \Delta (I) defines a morphism\textstyle _\Delta : \ {\mathbb {F... | {
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d60d321adb6eedcbb7070a3b08ed8b890accc03b | subsection | 20 | 144 | Duality | This means that we obtain a commutative diagram\textstyle {tikz/fig11}
\begin{}[row sep=0pt, column sep=50pt]
& \binom{E}{r} {dl}[swap]{j_{r,E}}{rrd}{\Delta } \\
{\mathbb {F}}_1^\pm \big [ \, x_I \, \big | \, I\in \binom{E}{r} \, \big ]\!\sslash \!\operatorname{{Pl}}(r,E) {ddd}[swap]{^{\mbox{[-8]$\vee $}}} {rrr}[swap]{... | {
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b38a1636913517435dd89c1778bd5477667e9023 | subsection | 21 | 144 | Rational point sets | In this section, we explain how the matroid space recovers classical objects like the Grassmannian, the Dressian and the MacPhersonian as rational point sets.Let B be a pasteurized ordered blueprint. By the universal property of the matroid space, \operatorname{Mat}(r,E)(B) corresponds to the set of B-matroids of rank ... | {
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d41bc5975c9ffcf8b977ccc7738a25807ccb503f | subsection | 22 | 144 | Rational point sets | It turns out that a topological hyperfield is the same as a topological pasture if identified with the associated pasture via the functor (-)^\textup {oblpr}:\operatorname{{HypFields}}\rightarrow \operatorname{{OBlpr}}^\pm . In the following, we consider the following topological pastures:the reals {\mathbb {R}} with t... | {
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7993aa64adfc6bbf5e31d06703a59fc60a347845 | subsection | 23 | 144 | Matroids | A matroid is the same as a {\mathbb {K}}-matroid where {\mathbb {K}}=\lbrace 0,1\rbrace \!\sslash \!\langle 0\leqslant 1+1,0\leqslant 1+1+1 \rangle is the Krasner hyperfield. Thus \operatorname{Mat}(r,E)({\mathbb {K}}) is the set of all matroids of rank r on E. The topology on {\mathbb {K}} turns \operatorname{Mat}(r,E... | {
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af7059436056433868fa4790ab5b0767753a7bbd | subsection | 24 | 144 | Oriented matroids and the MacPhersonian | Note that as a pasture, the sign hyperfield turns into {\mathbb {S}}=\lbrace 0,1,\rbrace \!\sslash \!{\mathcal {R}} where {\mathcal {R}} is generated by relations 0\leqslant 1+\cdots +1++\cdots + that contain at least one 1 and one .An oriented matroid is the same thing as a {\mathbb {S}}-matroid. Thus \operatorname{Ma... | {
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32ae49eaf0335bb8bea30e8a1d7a4a2d80762f15 | subsection | 25 | 144 | Subvector spaces and the Grassmannian | Let k be a field, which we identify with the pasture k^\bullet \!\sslash \!\langle 0\leqslant \sum a_i|\sum a_i=0\text{ in }k \rangle . (Note that this results from considering k as a partial field and applying the functor \operatorname{{PartFields}}\rightarrow \operatorname{{OBlpr}}^\pm or, equivalently, from consider... | {
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6b866d594a7379213b3f8af5d5cb6e92a5251b84 | subsection | 26 | 144 | The oriented matroid of real subvector spaces | The topology of {\mathbb {R}} endows \operatorname{Mat}(r,E)({\mathbb {R}}) with a topology that coincides with the usual topology of the real Grassmannian. The hyperfield morphism \textup {sign}:{\mathbb {R}}\rightarrow {\mathbb {S}} is continuous and therefore induces a continuous map\operatorname{Gr}(r,E)({\mathbb {... | {
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38d784f42d718a273ba986fc884f570424ed4b6e | subsection | 27 | 144 | Valuated matroids and the Dressian | A valuated matroid is the same thing as a -matroid, where is the tropical hyperfield. Thus \operatorname{Mat}(r,E)( is the set of all valuated matroids of rank r on E. An r-dimensional tropical linear space in {\mathbb {R}}^E is the geometric realization of a valuated matroid as a subspace of {\mathbb {R}}^E, analogous... | {
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591677cdf444912eb81b1ff3d2ab4a9c50e3867b | subsection | 28 | 144 | Regular matroids | It follows from our explanations in section that the subset of regular matroids in \operatorname{Mat}(r,E)({\mathbb {K}}) is equal to the image of the map \operatorname{Mat}(r,E)({\mathbb {F}}_1^\pm )\rightarrow \operatorname{Mat}(r,E)({\mathbb {K}}) induced by the unique morphism {\mathbb {F}}_1^\pm \rightarrow {\math... | {
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5ce81bb36ffd13f424da463d10922dc2d0007c1e | subsection | 29 | 144 | Realization spaces and the Tutte group | A new feature that comes along with the matroid space is the universal pasture associated with a matroid. We will introduce this notion and explain how it interacts with questions about the representability of matroids and realization spaces. We will
also discuss the analogous invariant for weak matroids and its relati... | {
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d91004f238b18ea892702fb2f64f800830105059 | subsection | 30 | 144 | The universal pasture of a matroid | We can associate with every matroid its universal pasture, which is derived from a certain residue field of the matroid space. We will define the universal pasture and describe its basic properties in this section.Let N=\#\binom{E}{r}-1. Recall from section REF that the matroid space comes with a closed immersion: \ \o... | {
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89a372d72c40699c7b5017f7a3638c65f3e4e0c8 | subsection | 31 | 144 | The universal pasture of a matroid | The stalk at {\mathfrak {p}}_{\mathcal {I}} is the ordered blueprint\textstyle {\mathcal {O}}_{\operatorname{Mat}(r,E),{\mathfrak {p}}_{\mathcal {I}}} \ = \ \big ({\mathbb {F}}_1^\pm \big [ \, x_I^\pm , x_J \, \big | \, I\in {\mathcal {I}},J\in {\mathcal {I}}^c \, \big ]\!\sslash \!\operatorname{{Pl}}(r,E)\big )_0where... | {
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45a04f4303e3041080c5bb343445d80cec8fb7f1 | subsection | 32 | 144 | The universal pasture of a matroid | The support of M is the image point x_M of _M and the universal pasture of M is k_M=k(x_M)^\pm .More explicitly, we have\textstyle k_M \ = \ \big ({\mathbb {F}}_1^\pm \big [ \, x_I^\pm \, \big | \, I\in {\mathcal {I}}\, \big ]\!\sslash \!\operatorname{{Pl}}(r,E)\big )_0^\pmwhere \operatorname{{Pl}}(r,E) is generated by... | {
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2a8b32ff609017479a4cea16307308b9f6a47b42 | subsection | 33 | 144 | The universal pasture of a matroid | This means that factors into a uniquely determined morphism \operatorname{Spec}{\mathbb {K}}\rightarrow \operatorname{Spec}k(x) followed by \operatorname{Spec}k(x)\rightarrow \operatorname{Mat}(r,E). This yields a morphism k(x)\rightarrow {\mathbb {K}}, which extends uniquely to a morphism k(x)^\pm \rightarrow {\mathbb... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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d579c98ad9b7ddc419214a2806afcc431958cc51 | subsection | 34 | 144 | The universal pasture of a matroid | Thus the morphism resulting from t_M^\ast with the canonical morphism \operatorname{Spec}k_M\rightarrow \operatorname{Mat}(r,E) must be equal to _M.Remark 2.6 As a consequence of Proposition REF and Corollary REF , we see that only the points x in the image of \Phi are supports of matroids.Lemma 2.7
Let k be a field a... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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3935c51f3369f2d9f0069fe0cb948dd17444a3c7 | subsection | 35 | 144 | The universal pasture of a matroid | By Lemma REF , this is equivalent to k(x)^\times =\lbrace 1,\rbrace , or k(x)={\mathbb {F}}_1^\pm , as claimed. Since {\mathbb {F}}_1^\pm is pasteurized and nonzero, Proposition REF implies that x is the support of a matroid.Example 2.10 (Support of the uniform matroid)
The uniform matroid of rank r on E is the matroi... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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71078d433251e749fef46b111055245e84d252a0 | subsection | 36 | 144 | The universal pasture of a matroid | Since the Plücker relations in the definition of \operatorname{Mat}(r,E) are merely inequalities, they do not identify any elements of the underlying monoid of {\mathbb {F}}_1^\pm \big [x_I \; \big | \; I\in \binom{E}{r}\big ]. As a result, the underlying topological space of \operatorname{Mat}(r,E) is the same as that... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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4a6a43071ea4b7a70b43d4302c47f822f50a4578 | subsection | 37 | 144 | The universal pasture of a matroid | The residue field at {\mathfrak {p}}_{\mathcal {I}} is\textstyle k({\mathfrak {p}}_{\mathcal {I}}) \ = \ {\mathcal {O}}_{\operatorname{Mat}(r,E),{\mathfrak {p}}_{\mathcal {I}}}\!\sslash \!\langle x_Jx_I^{-1}\equiv 0 \; | \; J\in {\mathcal {I}}^c \ranglewhere I\in {\mathcal {I}} is an arbitrary fixed index that allows u... | {
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"Matthew Baker",
"Oliver Lorscheid"
] | [
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3534a7d9eb7c0c9fe171ec4bb434345c9039f024 | subsection | 38 | 144 | The universal pasture of a matroid | The support of M is the image point x_M of _M and the universal pasture of M is k_M=k(x_M)^\pm .More explicitly, we have\textstyle k_M \ = \ \big ({\mathbb {F}}_1^\pm \big [ \, x_I^\pm \, \big | \, I\in {\mathcal {I}}\, \big ]\!\sslash \!\operatorname{{Pl}}(r,E)\big )_0^\pmwhere \operatorname{{Pl}}(r,E) is generated by... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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e1441f94409782a00b02f721a3f13c69febc8c98 | subsection | 39 | 144 | The universal pasture of a matroid | This means that factors into a uniquely determined morphism \operatorname{Spec}{\mathbb {K}}\rightarrow \operatorname{Spec}k(x) followed by \operatorname{Spec}k(x)\rightarrow \operatorname{Mat}(r,E). This yields a morphism k(x)\rightarrow {\mathbb {K}}, which extends uniquely to a morphism k(x)^\pm \rightarrow {\mathbb... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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cbc7be5172f09d151144b1338ec07e066c5f448f | subsection | 40 | 144 | The universal pasture of a matroid | Thus the morphism resulting from t_M^\ast with the canonical morphism \operatorname{Spec}k_M\rightarrow \operatorname{Mat}(r,E) must be equal to _M.Remark 2.6 As a consequence of Proposition REF and Corollary REF , we see that only the points x in the image of \Phi are supports of matroids.Lemma 2.7
Let k be a field a... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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4dd0007873c2ca940b5ec283fc28aa3666d9d8dd | subsection | 41 | 144 | The universal pasture of a matroid | By Lemma REF , this is equivalent to k(x)^\times =\lbrace 1,\rbrace , or k(x)={\mathbb {F}}_1^\pm , as claimed. Since {\mathbb {F}}_1^\pm is pasteurized and nonzero, Proposition REF implies that x is the support of a matroid.Example 2.10 (Support of the uniform matroid)
The uniform matroid of rank r on E is the matroi... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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fcdcbfca902697ebb412053b784970a59f25c892 | subsection | 42 | 144 | Universal pastures for rank 2-matroids on the four element set | In the following, we characterize the different universal pastures that can occur for \operatorname{Mat}(2,E) where E=\lbrace 1,2,3,4\rbrace . Note that \operatorname{Mat}(2,E) is defined by a single Plücker relation, namely0 \ \leqslant \ x_{1,2}x_{3,4} \ + \ \cdot x_{1,3}x_{2,4} \ + \ x_{1,4}x_{2,3}where we write x_{... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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1a1cca59c4b4f4da0375d0636517c70d88d038e0 | subsection | 43 | 144 | Case | By Corollary REF , we have k(x)^\pm =k(x)={\mathbb {F}}_1^\pm . In particular, x is the support of a matroid.By Lemma REF , we have x={\mathfrak {p}}_{\mathcal {I}} for a 2-subset {\mathcal {I}}=\lbrace I,J\rbrace of \binom{E}{r}. There are two cases. If I and J intersect nontrivially, thenk(x)^\pm \ = \ k(x) \ = \ {{\... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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b6e91981e9eb178345fbc7d7cc68bc1c6589a160 | subsection | 44 | 144 | Case | If K and L intersect nontrivially, then we havek(x) \ = \ \big ({{\mathbb {F}}_1^\pm }[x_{I}^{\pm 1},x_{J}^{\pm 1},x_{K}^{\pm 1},x_{L}^{\pm 1}]\!\sslash \!\langle 0\leqslant 1 \rangle \big )_0and k(x)^\pm =\lbrace 0\rbrace , i.e. x is not the support of a matroid. If K\cap L=\emptyset , thenk(x) \ = \ \big ({{\mathbb {... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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356375b7dbddc76de1bc76479928478fa606f5e4 | subsection | 45 | 144 | Case | As in the rank 4-case, we havek(x)^\pm \ = \ k(x)\!\sslash \!\langle \equiv ^ix_Kx_Lx_I^{-1}x_J^{-1} \rangle \ \simeq \ {{\mathbb {F}}_1^\pm }[x_{I}^{\pm 1},x_{J}^{\pm 1},x_{K}^{\pm 1},x_{N}^{\pm 1}]_0.Thus x is the support of a matroid.By Lemma REF , we have x={\mathfrak {p}}_{\mathcal {I}} for {\mathcal {I}}=\binom{E... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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a93714c1c91548a80872cc8cfb2aafa4e2cb3dac | subsection | 46 | 144 | Case | Thus we havek(x)^\pm \ = \ k(x) \ = \ {{\mathbb {F}}_1^\pm }[x_{I}^{\pm 1},x_{J}^{\pm 1},x_K^{\pm 1}]_0and x is the support of a matroid. If not—for instance, I\cap J=\emptyset — thenk(x) \ = \ \big ({{\mathbb {F}}_1^\pm }[x_{I}^{\pm 1},x_{J}^{\pm 1},x_{K}^{\pm 1}]\!\sslash \!\langle 0\leqslant 1 \rangle \big )_0,as in... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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74cdbf46ef5ad07d005d9811bf65cf11d8eb25b5 | subsection | 47 | 144 | Case | Then we havek(x) \ = \ \big ({{\mathbb {F}}_1^\pm }[x_{I}^{\pm 1},x_{J}^{\pm 1},x_{K}^{\pm 1},x_{L}^{\pm 1},x_{N}^{\pm 1}]\!\sslash \!\langle 0\leqslant x_Ix_J+^ix_Kx_L \rangle \big )_0where i=0 or 1, depending on I, J, K and L. As in the rank 4-case, we havek(x)^\pm \ = \ k(x)\!\sslash \!\langle \equiv ^ix_Kx_Lx_I^{-1... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
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019f58a4d6ad4f5a6ab7a4c8bb19a867fbbb0bd2 | subsection | 48 | 144 | Realization spaces | Let k be a field. The realization space of a matroid M is the subset of the Grassmannian over k that consists of the subvector spaces whose associated matroid is M. These realization spaces have been used for proving that several moduli spaces, such as Hilbert schemes and moduli spaces of curves, can become arbitrarily... | {
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"raw": "Ravi Vakil. Murphy's law in algebraic geometry: badly-behaved deformation spaces. Invent. Math., 164(3):569–590, 2006.",
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"Matthew Baker",
"Oliver Lorscheid"
] | [
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76de6d8ebe060a76d0a28bd568d4944b228b5445 | subsection | 49 | 144 | Realization spaces | In other words, \operatorname{Spec}k_M is the fine moduli space of realization spaces for M. In down-to-earth terms, this means the following:Theorem 2.14
Let M be a matroid and _M:\operatorname{Spec}k_M\rightarrow \operatorname{Mat}(r,E) the inclusion of the universal pasture k_M of M into the matroid space. Let F be... | {
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"Matthew Baker",
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71e741c1f15f8027b17711d977e9061808068bd0 | subsection | 50 | 144 | Realization spaces | In this section, we show that realization spaces are the same as morphism sets from universal pastures.Let \Delta :\binom{E}{r}\rightarrow {\mathbb {K}} be a Grassmann-Plücker function, M=[\Delta ] the corresponding matroid and _M:{\mathbb {K}}\rightarrow \operatorname{Mat}(r,E) its characteristic morphism. Let F be a ... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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deabee60110f6b8e5470900905cc2727b20916c3 | subsection | 51 | 144 | Realization spaces | Let F be a pasture. The map\begin{array}{cccc}
_{M,\ast }: & \operatorname{Hom}(k_M,F) & \longrightarrow & {\mathcal {X}}_M(F) \\
& f & \longmapsto & _M\circ f^\ast \end{array}is a bijection that is functorial in F.We will show that every morphism :\operatorname{Spec}F\rightarrow \operatorname{Mat}(r,E) in {\mathcal {X... | {
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"Matthew Baker",
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cddb93ce1d157168aecbe3364537bc686f4ff03f | subsection | 52 | 144 | The weak matroid space | There is a variant of the matroid space for weak matroids, which leads to the notion of the weak universal pasture of a matroid. Although it turns out that the weak matroid space is not a moduli space for weak matroids, it turns out that the universal pasture is a very useful object for matroid theory for its connectio... | {
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"Oliver Lorscheid"
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7510118934a4e55fa5ec36720cfe38f390f6f57f | subsection | 53 | 144 | The weak matroid space | This justifies our abuse of terminology.Definition 2.17 The weak matroid space of rank r on E is the ordered blue scheme\textstyle \operatorname{Mat}^w(r,E) \quad = \quad \operatorname{Proj}\Big ( \, {\mathbb {F}}_1^\pm \big [ \, x_I \, \big | \, I\in \binom{E}{r} \, \big ]\!\sslash \!\operatorname{{Pl}}^w(r,E) \, \Big... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
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b98bcabd5fb8238b90d06997d29fc8b0a0cbbd68 | subsection | 54 | 144 | The weak matroid space | The weak characteristic morphism is the morphism _M^w=^w\circ _M:\operatorname{Spec}{\mathbb {K}}\rightarrow \operatorname{Mat}^w(r,E). The weak support of M is the image x_M^w=^w(x_M) of x_M in \operatorname{Mat}^w(r,E). The weak universal pasture of M is the pasteurization k_M^w=k(x_M^w)^\pm of the residue field k(x^... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
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3494bb9a0ac1ae2fd25e7426ef64aaae377468ab | subsection | 55 | 144 | The weak matroid space | Then the map\begin{array}{cccc}
_{M,\ast }^w: & \operatorname{Hom}(k_M^w,F) & \longrightarrow & {\mathcal {X}}_M^w(F) \\
& f & \longmapsto & _M^w\circ f^\ast \end{array}is a bijection.The proof is analogous to that of the corresponding result for strong F-matroids, see Theorem REF . For completeness, we outline the ide... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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975765f60bdd073367ba8a05e40828f32d3c70a6 | subsection | 56 | 144 | The weak matroid space | Consider the 3-term Plücker relation0 \ \leqslant \ \Delta (I_{1,2}) \, \Delta (I_{3,4}) \ + \ \, \Delta (I_{1,3}) \, \Delta (I_{2,4}) \ + \ \Delta (I_{1,4}) \, \Delta (I_{2,3})for I=\lbrace i_0\rbrace and i_1<i_2<i_3<i_4 with i_1,i_2,i_3,i_4\notin \lbrace i_0\rbrace , where I_{k,l}=\lbrace i_0,i_k,i_l\rbrace . In orde... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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4736a34e3632fdf4360c0bfcb3ebfbe37bb12d95 | subsection | 57 | 144 | The weak matroid space | Therefore \Delta is not a weak Grassmann-Plücker function.Let {\mathcal {I}} be the complement of \lbrace J,J^c\rbrace in \binom{E}{r} and x^w={\mathfrak {p}}_{\mathcal {I}} the corresponding point of the weak matroid space \operatorname{Mat}^w(3,E). Since all 3-term Plücker relations for \Delta are trivial, the residu... | {
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"Matthew Baker",
"Oliver Lorscheid"
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5ef87b6e43fd6c077e3fac28a436c6942f7152e4 | subsection | 58 | 144 | The weak matroid space | For instance, the phase hyperfield, which is the hyperfield quotient of by {\mathbb {R}}^\times , admits weak matroids that are not strong. See Example 2.36 in for details.In the following, we shall call a pasture F perfect if the associated tract F^\textup {tract} is perfect. Since the matroid theories of F and F^\tex... | {
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"Matthew Baker",
"Oliver Lorscheid"
] | [
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16ba7d0d76b7de7bc86866d7ebc2520633f671f5 | subsection | 59 | 144 | The weak matroid space | Let {\mathcal {L}}^w_\textup {univ}=^\ast ({\mathcal {O}}(1)) be the pullback of the tautological bundle {\mathcal {O}}(1) on {\mathbb {P}}^N_{{\mathbb {F}}_1^\pm } to \operatorname{Mat}^w(r,E).The identity map induces a morphism\textstyle {\mathbb {F}}_1^\pm \big [ \, x_I \, \big | \, I\in \binom{E}{r} \, \big ]\!\ssl... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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07bfc6201b9af1d93da92cbce78d8bf9e5bc2dad | subsection | 60 | 144 | The weak matroid space | The weak realization space of M over F is the set{\mathcal {X}}_M^w(F) \ = \ \big \lbrace \, :\operatorname{Spec}F\rightarrow \operatorname{Mat}^w(r,E) \, \big | \, \circ t_F^\ast =_M^w \, \big \rbraceof all weak F-matroids that represent M.Recall from Remark REF the definition of a perfect pasture.Lemma 2.20
Let M be... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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1604c10bc864d1ead72013654bfe6da042f9ff6d | subsection | 61 | 144 | The weak matroid space | Since the locus of points of \operatorname{Mat}^w(r,E) supporting matroids is not locally closed, but merely constructible in general, this locus does not inherit a scheme structure from \operatorname{Mat}^w(r,E) in an obvious way.For instance, it is a well-known fact that the 3-term Plücker relations do not suffice, i... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
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d6b735728289dbef5dd41a534a7c1ebf60716440 | subsection | 62 | 144 | The weak matroid space | The Plücker relation for I and I^{\prime } is0 \ \leqslant \ \sum _{k=1}^4 ^k \cdot \overline{\Delta }(I\cup \lbrace j_k\rbrace ) \cdot \overline{\Delta }(I^{\prime }-\lbrace j_k\rbrace )where =1. The sum on the right hand side has precisely one nonzero term, namely\overline{\Delta }(I\cup \lbrace j_l\rbrace ) \cdot \o... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
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c55aa9fa6ec512727fd7fafff5546e7a676742e3 | subsection | 63 | 144 | The Tutte group | The Tutte group is introduced by Dress and Wenzel in and used as a tool to study the representability of matroids and to provide cryptomorphisms for matroids over fuzzy rings, cf. . In this section, we show that the Tutte group is precisely the unit group of the weak universal pasture.For the following characterization... | {
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"Matthew Baker",
"Oliver Lorscheid"
] | [
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7255b049f2d82173097cec4375ead62ccdec738d | subsection | 64 | 144 | The Tutte group | As explained in the proof of Lemma REF , we have x_M={\mathfrak {p}}_{{\mathcal {B}}^c} where {\mathcal {B}}^c is the complement of {\mathcal {B}} in \binom{E}{r}, andk(x_M^w)^\times \ = \ k(x_M)^\times \ = \ \Big \lbrace \, ^i \, \cdot \, \prod _{I\in {\mathcal {B}}} \ x_I^{e_I} \, \Big | \, i\in \lbrace 0,1\rbrace ,e... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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bda0318efae8089da0d91d61b1cdbf5c99f5128f | subsection | 65 | 144 | The Tutte group | To enable ourselves to work with degree-0 elements, we will work with two graded abelian groups G_M^{\prime } and H_M, which contain M and (k_M^w)^\times , respectively, as subquotients.Namely, we define G_M^{\prime } as the abelian group generated by the symbols and X_{(i_1,\cdots ,i_r)} for every (i_1,\cdots ,i_r)\in... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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1e1f9f27a1839fb395fcba9a090488cb71676450 | subsection | 66 | 144 | The Tutte group | Thus the theorem follows if we can show that the isomorphism f identifies \ker h with \ker g.As our next step, we exhibit a set of generators for \ker h. The kernel of h consists of all weak inverses of 1 in k(x_M^w). Such elements must come from the 3-term Plücker relation0 \leqslant x_{I,1,2} \, x_{I,3,4} \ + \ \, x_... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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a9cc5e57df010a3063feb8e21c5165959e0cdb6b | subsection | 67 | 144 | The Tutte group | In the following, we will show that f maps this set to the set of generators for \ker g that we used in the definition of G_M=G_M^{\prime }/\ker g.Let j_1,\cdots ,j_{r-2},k_1,k_2,l_1,l_2 be pairwise different elements of E, and define I=\lbrace j_1,\cdots ,j_{r-2}\rbrace and \lbrace i_1,i_2,i_3,i_4\rbrace =\lbrace k_1,... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0e0c6e4333a290fb6b8c82b8f3efd21b50c1191e | subsection | 68 | 144 | The Tutte group | For instance, if j_1<\cdots <j_{r-2}, then we haveN \ = \ \sum _{i\in \lbrace k_1,k_2,l_1,l_2\rbrace } \#\lbrace j\in I|i<j\rbrace .From these considerations, we obtain the equality\frac{X_{(j_1,\cdots ,j_{r-2},k_p,l_q)} \, X_{(j_1,\cdots ,j_{r-2},k_{p^{\prime }},l_{q^{\prime }})}}{X_{(j_1,\cdots ,j_{r-2},k_p,l_{q^{\pr... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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f9a687dfc8cb308d96802006eeede72f61bbe255 | subsection | 69 | 144 | The Tutte group | We obtain\frac{X_{(j_1,\cdots ,j_{r-2},k_1,l_1)} \, X_{(j_1,\cdots ,j_{r-2},k_{2},l_{2})}}{X_{(j_1,\cdots ,j_{r-2},k_1,l_{2})} \, X_{(j_1,\cdots ,j_{r-2},k_{2},l_{1})}} \quad = \quad \frac{X_{I\cup \lbrace i_1,i_3\rbrace } \, X_{I\cup \lbrace i_2,i_4\rbrace }}{X_{I\cup \lbrace i_1,i_4\rbrace } \, X_{I\cup \lbrace i_2,i... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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7cc40d1d9313df4df0faa8c880e967ebd2d466f7 | subsection | 70 | 144 | The Tutte group | Since ^2=1, we have\frac{X_{(j_1,\cdots ,j_{r-2},k_1,l_1)} \, X_{(j_1,\cdots ,j_{r-2},k_{2},l_{2})}}{X_{(j_1,\cdots ,j_{r-2},k_1,l_{2})} \, X_{(j_1,\cdots ,j_{r-2},k_{2},l_{1})}} \quad = \quad \frac{X_{I\cup \lbrace i_1,i_2\rbrace } \, X_{I\cup \lbrace i_3,i_4\rbrace }}{X_{I\cup \lbrace i_1,i_3\rbrace } \, X_{I\cup \lb... | {
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"Matthew Baker",
"Oliver Lorscheid"
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7dda119fa0b879d2583f1d903ca162b44b32fc97 | subsection | 71 | 144 | The Tutte group | The Tutte group M of M is defined as the subgroup of G_M that is generated by and elements of the form X_{(i_1,\cdots ,i_r)}X_{(j_1,\cdots ,j_r)}^{-1} with \lbrace i_1,\cdots ,i_r\rbrace ,\lbrace j_1,\cdots ,j_r\rbrace \in {\mathcal {B}}.Let x_M be the support of M and x_M^w its weak support. The natural map k(x_M^w)\r... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
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68e8ea827d51a4b3334b2df72dbac8569ea67502 | subsection | 72 | 144 | The Tutte group | To enable ourselves to work with degree-0 elements, we will work with two graded abelian groups G_M^{\prime } and H_M, which contain M and (k_M^w)^\times , respectively, as subquotients.Namely, we define G_M^{\prime } as the abelian group generated by the symbols and X_{(i_1,\cdots ,i_r)} for every (i_1,\cdots ,i_r)\in... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
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30ee6e2bb4a996cef6853deb0937968076d4670d | subsection | 73 | 144 | The Tutte group | Thus the theorem follows if we can show that the isomorphism f identifies \ker h with \ker g.As our next step, we exhibit a set of generators for \ker h. The kernel of h consists of all weak inverses of 1 in k(x_M^w). Such elements must come from the 3-term Plücker relation0 \leqslant x_{I,1,2} \, x_{I,3,4} \ + \ \, x_... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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ff078215ae19161e569d6613127179c67df7fe14 | subsection | 74 | 144 | The Tutte group | In the following, we will show that f maps this set to the set of generators for \ker g that we used in the definition of G_M=G_M^{\prime }/\ker g.Let j_1,\cdots ,j_{r-2},k_1,k_2,l_1,l_2 be pairwise different elements of E, and define I=\lbrace j_1,\cdots ,j_{r-2}\rbrace and \lbrace i_1,i_2,i_3,i_4\rbrace =\lbrace k_1,... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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b70f5c17d254d0011ddbae213ad13d43e378458e | subsection | 75 | 144 | The Tutte group | For instance, if j_1<\cdots <j_{r-2}, then we haveN \ = \ \sum _{i\in \lbrace k_1,k_2,l_1,l_2\rbrace } \#\lbrace j\in I|i<j\rbrace .From these considerations, we obtain the equality\frac{X_{(j_1,\cdots ,j_{r-2},k_p,l_q)} \, X_{(j_1,\cdots ,j_{r-2},k_{p^{\prime }},l_{q^{\prime }})}}{X_{(j_1,\cdots ,j_{r-2},k_p,l_{q^{\pr... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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7bc4bddedd5aebb00337080d6227af3071f31f2b | subsection | 76 | 144 | The Tutte group | We obtain\frac{X_{(j_1,\cdots ,j_{r-2},k_1,l_1)} \, X_{(j_1,\cdots ,j_{r-2},k_{2},l_{2})}}{X_{(j_1,\cdots ,j_{r-2},k_1,l_{2})} \, X_{(j_1,\cdots ,j_{r-2},k_{2},l_{1})}} \quad = \quad \frac{X_{I\cup \lbrace i_1,i_3\rbrace } \, X_{I\cup \lbrace i_2,i_4\rbrace }}{X_{I\cup \lbrace i_1,i_4\rbrace } \, X_{I\cup \lbrace i_2,i... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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bb0849b898280fc77d18b0151af59f106f40d4d1 | subsection | 77 | 144 | The Tutte group | Since ^2=1, we have\frac{X_{(j_1,\cdots ,j_{r-2},k_1,l_1)} \, X_{(j_1,\cdots ,j_{r-2},k_{2},l_{2})}}{X_{(j_1,\cdots ,j_{r-2},k_1,l_{2})} \, X_{(j_1,\cdots ,j_{r-2},k_{2},l_{1})}} \quad = \quad \frac{X_{I\cup \lbrace i_1,i_2\rbrace } \, X_{I\cup \lbrace i_3,i_4\rbrace }}{X_{I\cup \lbrace i_1,i_3\rbrace } \, X_{I\cup \lb... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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902c24c8b5c6b4ad8d1d92a5be3cb5cb9272fdbc | subsection | 78 | 144 | Cross ratios and rescaling classes | In this section, we will define and study the properties of the foundation of a matroid, which is a subpasture of the weak universal pasture that is closely related to the inner Tutte group from and the universal partial field from .The key notions in this section are cross ratios, rescaling classes, fundamental elemen... | {
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"Matthew Baker",
"Oliver Lorscheid"
] | [
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c1362ed645fb0b773beba8250953101c2b44f82c | subsection | 79 | 144 | Cross ratios | The study of cross ratios of four points on a line belongs to the oldest themes in mathematics and finds its earliest traces in the writings of Pappus of Alexandria (). Its main property is that it is invariant under projective transformation and that it characterizes the ratios of the pairwise differences between the ... | {
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"Matthew Baker",
"Oliver Lorscheid"
] | [
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90d85ecb859e639abf4a9ad79c65f093ab84c4a8 | subsection | 80 | 144 | Cross ratios | Then the cross ratio satisfies the relations\operatorname{Cr}_\Delta \big (.{\mathcal {I}}\big ) \ = \ \operatorname{Cr}_\Delta ({\mathcal {I}}) \qquad \text{and} \qquad \operatorname{Cr}_\Delta \big (.{\mathcal {I}}\big ) \ = \ \operatorname{Cr}_\Delta ({\mathcal {I}})^{-1}for every in the Klein four group V=\big \lbr... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
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052c5a58c6164e0cf5f6572592e88d3bf3c9b468 | subsection | 81 | 144 | Cross ratios | Thus \Delta satisfies the 3-term Grassmann-Plücker relation0 \ \leqslant \ \Delta _{I,(1),(2)} \ \Delta _{I,(3),(4)} \ + \ \ \Delta _{I,(1),(3)} \ \Delta _{I,(2),(4)} \ + \ \Delta _{I,(1),(4)} \ \Delta _{I,(2),(3)}where \Delta _{I,k,l}=\Delta \big (I\cup \lbrace i_k,i_l\rbrace \big ) and is the permutation determined b... | {
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"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
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34d42a2ddc3975a31e98401b62d0934016d8a6ed | subsection | 82 | 144 | Cross ratios | For more details on the developments of cross ratios in general and explanations of their relevance for matroid theory, we refer to the book of Richter-Gebert.Let F be a pasture and M a matroid of rank r on E. The cross ratios of M in F are indexed by certain quadrangles or 4-cycles in the basis exchange graph of M.We ... | {
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"Matthew Baker",
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] | [
"math.AG"
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c039b7a94dc8c58cbbf786adabdb120603a3a578 | subsection | 83 | 144 | Cross ratios | If {\mathcal {I}} is non-degenerate, then \operatorname{Cr}_\Delta (.{\mathcal {I}}) is defined for all permutations and we have the identity\operatorname{Cr}_\Delta ({\mathcal {I}}) \ \cdot \ \operatorname{Cr}_\Delta \big ((123).{\mathcal {I}}\big ) \ \cdot \ \operatorname{Cr}_\Delta \big ((321).{\mathcal {I}}\big ) \... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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6e0b44a8325fb7771fea8487903d0354bf0e4d21 | subsection | 84 | 144 | Cross ratios | After dividing by \Delta _{I,1,4}\Delta _{I,2,3}, we conclude that \operatorname{Cr}_\Delta ({\mathcal {I}})\in \lbrace 1,\rbrace , as claimed.If {\mathcal {I}} is non-degenerate, then all three terms are nonzero. After multiplying by appropriate powers of , this yields0 \ \leqslant \ \Delta _{I,1,2} \ \Delta _{3,4} \ ... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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b463a33526652bca1ab2d946ccea7c1d8e395421 | subsection | 85 | 144 | Foundations | Pendavingh and van Zwam exhibit in the role of fundamental elements for the representability of matroids over partial fields. In this section, we extend this concept to pasteurized ordered blueprints, which makes this theory applicable to matroids over all pastures. Since there is a discrepancy between the signs of cro... | {
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692781ed277ea620d2dd9c3d113f90ccb376fad3 | subsection | 86 | 144 | Foundations | The foundation of M is the subpasture k_M^f=(k_M^w)^\textup {found} of k_M^w.Let \Delta :\binom{E}{r}\rightarrow k_M^w be the weak Grassmann-Plücker function with \Delta (I)=x_I/x_{I_0} for some fixed basis I_0 of M. The universal cross ratio function of M is the function \operatorname{Cr}_M^\textup {univ}:\Omega _M\ri... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
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"math.AG"
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84d25fb641726df5acabbc667b94f06943a9d5b5 | subsection | 87 | 144 | Foundations | Theorem REF .In case of the uniform matroid M, it is easily seen that its foundation isk_M^f \ = \ {{\mathbb {F}}_1^\pm }[T_1^\pm ,T_2^\pm ]\!\sslash \!\langle 0\leqslant T_1+T_2+1 \ranglewhere T_1 and T_2 stand for the cross ratiosT_1 \ = \ \frac{x_{1,2}x_{3,4}}{x_{1,4}x_{2,3}} \quad \text{and} \quad T_2 \ = \ \frac{x... | {
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fafcfd66e9d0591e2474f6d5d171a866dc7104c6 | subsection | 88 | 144 | Foundations | An effective upper bound for the number of fundamental elements, depending only on the rank of R^\times as an abelian group, is given in . There is a similar result for characteristic p>0 if one counts solutions up to p-th powers; cf. .The relevance of fundamental elements and foundations for matroid theory is that the... | {
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"Matthew Baker",
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5a407bc9299b52023411077cf7db2334438367d9 | subsection | 89 | 144 | Foundations | Recall that there is a unique Plücker relation in this case, which is0 \ \leqslant \ x_{1,2}x_{3,4} \ + \ \cdot x_{1,3}x_{2,4} \ + \ x_{1,4}x_{2,3}where we write x_{i,j} for x_{\lbrace i,j\rbrace }. If any of the three terms is zero, then all cross ratios are 1 or by Lemma REF and thus k_M^f={{\mathbb {F}}_1^\pm }. Fro... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
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f22ccb30ae3d78012ea35ef83eb4e2f7389e3e99 | subsection | 90 | 144 | The inner Tutte group | Let M be a matroid of rank r on E and {\mathcal {B}} the set of bases of M. As a consequence of Theorem REF , the Tutte group M of M is isomorphic to the abelian group generated by and \prod _{I\in {\mathcal {B}}} X_I^{e_I} with \sum e_I=0 modulo the relations ^2=1 and\frac{X_{I,1,2}\ X_{I,3,4}}{X_{I,2,3}\ X_{I,4,1}} \... | {
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fcd9b2d1c7fdef0623b88b992f8b7de7ae6fac0b | subsection | 91 | 144 | The inner Tutte group | Recall from Theorem REF that the association \prod x_I^{e_I}\mapsto \prod X_I^{e_I} defines an isomorphism (k_M^w)^\times \rightarrow M between the units of the weak universal pasture and the Tutte group of M.Corollary 3.11
The isomorphism (k_M^w)^\times \rightarrow M restricts to an isomorphism (k_M^f)^\times \righta... | {
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} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
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1d61119d19586202fb1a366232bd11fa7e8e8fd1 | subsection | 92 | 144 | The inner Tutte group | By the very construction of B[T_1^{\pm 1},\cdots ,T_s^{\pm 1}], we have B[T_1^{\pm 1},\cdots ,T_s^{\pm 1}]^\times \simeq B^\times \times {\mathbb {Z}}^s.Corollary 3.12
The weak universal pasture k_M^w is isomorphic to k_M^f[T_1^{\pm 1},\cdots ,T_s^{\pm 1}] for some s\geqslant 0.The additive relations of k_M^w are gene... | {
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"Oliver Lorscheid"
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f6df59df67f170b2317b315f957e24d9d83f4783 | subsection | 93 | 144 | The inner Tutte group | Composing it with the inclusion k_M^f\rightarrow k_M^w yields the desired morphism k_M^f\rightarrow F. Thus (REF )\Rightarrow (REF ).If there is a morphism k_M^f\rightarrow F, then its image is contained in F^\textup {found}. By Corollary REF , we have k_M^w\simeq k_M^f[T_1^{\pm 1},\cdots ,T_s^{\pm 1}], and thus there ... | {
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"Oliver Lorscheid"
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a0723add37aa124c947420816518f7b5efee280a | subsection | 94 | 144 | The inner Tutte group | Its proof is relies on Tutte's “fundamental theorem on linear subclasses” (Theorem 4.34 in ), which is significantly easier to prove than the relatively deeper parts of Tutte's homotopy theory for matroids found in and (and also exposited in ).Theorem 3.10
The inner Tutte group M^{(0)} is generated by and the elements... | {
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015773e550d2d739f680f2f5a573b72b4b627b9d | subsection | 95 | 144 | The inner Tutte group | In particular, the ordered blueprint B[T_1^{\pm 1},\cdots ,T_s^{\pm 1}] is pasteurized if and only if B is so. By the very construction of B[T_1^{\pm 1},\cdots ,T_s^{\pm 1}], we have B[T_1^{\pm 1},\cdots ,T_s^{\pm 1}]^\times \simeq B^\times \times {\mathbb {Z}}^s.Corollary 3.12
The weak universal pasture k_M^w is isom... | {
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870351d69fccc870551a6295c448048c69ca1d50 | subsection | 96 | 144 | The inner Tutte group | Thus (REF )\Rightarrow (REF ).If M is weakly representable over F, then there exists a morphism k_M^w\rightarrow F by Proposition REF . Composing it with the inclusion k_M^f\rightarrow k_M^w yields the desired morphism k_M^f\rightarrow F. Thus (REF )\Rightarrow (REF ).If there is a morphism k_M^f\rightarrow F, then its... | {
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} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
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e31dfaa77d1aad0a8645fcf61dcbbad93fcd39f9 | subsection | 97 | 144 | Rescaling classes | Let B be a pasteurized ordered blueprint and T(B) the set of functions t:E\rightarrow B^\times , which comes with the structure of an abelian group with respect to the product t\cdot t^{\prime }(i)=t(i)\cdot t^{\prime }(i). For a subset I of E, we define t_I=\prod _{i\in I}t(i). For t\in T(B) and a weak Grassmann-Plück... | {
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... | 1809.03542 | The moduli space of matroids | [
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14f28e68d5eb9365550da39ffea0cbaaa126d528 | subsection | 98 | 144 | Rescaling classes | Equivalence of representations A and A^{\prime }, in the sense of , is defined in terms of realizations as incidence geometries inside {\mathbb {P}}^{r-1}(k): A and A^{\prime } are said to be equivalent if there is an automorphism of {\mathbb {P}}^{r-1}(k) (as an incidence geometry) which identifies the respective real... | {
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... | 1809.03542 | The moduli space of matroids | [
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8238b7af32512d4819a85aff5337814116b0ed9c | subsection | 99 | 144 | Rescaling classes | The action of T(B) on \operatorname{Mat}(r,E)(B) results from applying \operatorname{Hom}(\operatorname{Spec}B,-) to the morphism T\times \operatorname{Mat}(r,E)\rightarrow \operatorname{Mat}(r,E).Note that a morphism f:F\rightarrow F^{\prime } of pastures induces a map f_\ast :{\mathcal {X}}^w(F)\rightarrow {\mathcal ... | {
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} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
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84e0769f186f70b20fb3952287e226137f2a968f | subsection | 100 | 144 | Rescaling classes | Then for I_{k,l}=I\cup \lbrace i_k,i_l\rbrace , we have t_{I_{1,2}}t_{I_{3,4}}=t_{I_{2,3}}t_{I_{4,1}} and thus\operatorname{Cr}_{\Delta ^{\prime }}({\mathcal {I}}) \quad = \quad \frac{\Delta ^{\prime }_{I,1,2}\Delta ^{\prime }_{I,3,4}}{\Delta ^{\prime }_{I,2,3}\Delta ^{\prime }_{I,4,3}} \quad = \quad \frac{at_{I_{1,2}}... | {
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05a6d2ddbe761fc65585234d644e69205b6b224d | subsection | 101 | 144 | Rescaling classes | This establishes the claim.Thus the morphism \deg _E:M\rightarrow {\mathbb {Z}}^E is the direct sum of its restrictions to the support of the connected components of M and the rank of the cokernel is the sum of the ranks for each summand. Therefore we may assume that M is connected, and we are left to show that under t... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
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1fa63e94b4b26133dae5ed9b784e7f6283473060 | subsection | 102 | 144 | Rescaling classes | Since _{[\Delta ]} sends \prod x_I^{e_I} to \prod \Delta (I)^{e_I}, and similarly for _{[\Delta ^{\prime }]}, we have \operatorname{Cr}_\Delta =_{[\Delta ]}\circ \operatorname{Cr}_M^\textup {univ} and \operatorname{Cr}_{\Delta ^{\prime }}=_{[\Delta ^{\prime }]}\circ \operatorname{Cr}_M^\textup {univ}. Thus for every {\... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
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d5d95312464847465e1b57ec6cbb80d90098a565 | subsection | 103 | 144 | Rescaling classes | Then we have\textstyle \prod \Delta (I)^{e_I} \ = \ _{[\Delta ]}\big ( \prod x_I^{e_I}\big ) \ = \ t\big (\sum e_I_I\big ) \ _{[\Delta ^{\prime }]}\big ( \prod x_I^{e_I}\big ) \ = \ \prod t_I^{e_I} \, \cdot \, \prod \Delta ^{\prime }(x_I)^{e_I} \ = \ (t.\Delta ^{\prime })(I)^{e_I}for all e_I with \sum e_I=0, where I va... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
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11e56a246390367313a00df4774d3616bcbaa244 | subsection | 104 | 144 | Rescaling classes | Then the map\begin{array}{cccc}
\Phi : & {\mathcal {X}}_M^f(F) & \longrightarrow & \operatorname{Hom}(k_M^f,F) \\
& [\Delta ] & \longmapsto & _{[\Delta ]}\vert _{k_M^f}
\end{array}is a bijection that is functorial in F.By Theorem REF , two Grassmann-Plücker functions \Delta and \Delta ^{\prime } that represent M are re... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
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"math.AG"
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1d1ee2e7bbe0aa81cc4d2e1a06969df7094eb777 | subsection | 105 | 144 | Rescaling classes | Since the image of _{[\Delta ]}^f is contained in F^\textup {found}, the rescaling class of [\Delta ] comes from a class of an F^\textup {found}-matroid [\Delta ^{\prime }] that is represented by a Grassmann-Plücker function \Delta ^{\prime }:\binom{E}{r}\rightarrow F^\textup {found}.Thus \Delta and \circ \Delta ^{\pri... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
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43067ed2c88c09a11ac2c46a2189ab8e93a578d3 | subsection | 106 | 144 | Rescaling classes | Thus there exists a GIT quotient X^f_M=X^w_M/T^{\prime } as a scheme, and this quotient satisfies X^f_M(k)={\mathcal {X}}^f_M(k) for every field k.Let B be a pasteurized ordered blueprint and T(B) the set of functions t:E\rightarrow B^\times , which comes with the structure of an abelian group with respect to the produ... | {
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c1d7bf40cce8ba5c8cfe839e434c8a643c7fa957 | subsection | 107 | 144 | Rescaling classes | Equivalence of representations A and A^{\prime }, in the sense of , is defined in terms of realizations as incidence geometries inside {\mathbb {P}}^{r-1}(k): A and A^{\prime } are said to be equivalent if there is an automorphism of {\mathbb {P}}^{r-1}(k) (as an incidence geometry) which identifies the respective real... | {
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"Matthew Baker",
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2869180548326fcff87b3b3b52fe4981cdbe97a3 | subsection | 108 | 144 | Rescaling classes | The action of T(B) on \operatorname{Mat}(r,E)(B) results from applying \operatorname{Hom}(\operatorname{Spec}B,-) to the morphism T\times \operatorname{Mat}(r,E)\rightarrow \operatorname{Mat}(r,E).Note that a morphism f:F\rightarrow F^{\prime } of pastures induces a map f_\ast :{\mathcal {X}}^w(F)\rightarrow {\mathcal ... | {
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} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
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e0374285ea2df2fbdf433c7ad58a1c14c99368d0 | subsection | 109 | 144 | Rescaling classes | Then for I_{k,l}=I\cup \lbrace i_k,i_l\rbrace , we have t_{I_{1,2}}t_{I_{3,4}}=t_{I_{2,3}}t_{I_{4,1}} and thus\operatorname{Cr}_{\Delta ^{\prime }}({\mathcal {I}}) \quad = \quad \frac{\Delta ^{\prime }_{I,1,2}\Delta ^{\prime }_{I,3,4}}{\Delta ^{\prime }_{I,2,3}\Delta ^{\prime }_{I,4,3}} \quad = \quad \frac{at_{I_{1,2}}... | {
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"arxiv_id": "",
"doi": "10.1016/0001-8708(89)90013-3",
"end": 991,
"openalex_id": "https://openalex.org/W2086795154",
"raw": "Andreas W. M. Dress and Walter Wenzel. Geometric algebra for combinatorial geometries. Adv. Math., 77(1):1–36, 1989.",
"source_ref_i... | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
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86d64272743d66741ddbad7d23b53e809d42209b | subsection | 110 | 144 | Rescaling classes | This establishes the claim.Thus the morphism \deg _E:M\rightarrow {\mathbb {Z}}^E is the direct sum of its restrictions to the support of the connected components of M and the rank of the cokernel is the sum of the ranks for each summand. Therefore we may assume that M is connected, and we are left to show that under t... | {
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"Matthew Baker",
"Oliver Lorscheid"
] | [
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2179daaeef6e799e4126392fc75836c80ddbde8e | subsection | 111 | 144 | Rescaling classes | Since _{[\Delta ]} sends \prod x_I^{e_I} to \prod \Delta (I)^{e_I}, and similarly for _{[\Delta ^{\prime }]}, we have \operatorname{Cr}_\Delta =_{[\Delta ]}\circ \operatorname{Cr}_M^\textup {univ} and \operatorname{Cr}_{\Delta ^{\prime }}=_{[\Delta ^{\prime }]}\circ \operatorname{Cr}_M^\textup {univ}. Thus for every {\... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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4739920bfa31161ff8cb881687f3a1afe55b570a | subsection | 112 | 144 | Rescaling classes | Then we have\textstyle \prod \Delta (I)^{e_I} \ = \ _{[\Delta ]}\big ( \prod x_I^{e_I}\big ) \ = \ t\big (\sum e_I_I\big ) \ _{[\Delta ^{\prime }]}\big ( \prod x_I^{e_I}\big ) \ = \ \prod t_I^{e_I} \, \cdot \, \prod \Delta ^{\prime }(x_I)^{e_I} \ = \ (t.\Delta ^{\prime })(I)^{e_I}for all e_I with \sum e_I=0, where I va... | {
"cite_spans": []
} | 1809.03542 | The moduli space of matroids | [
"Matthew Baker",
"Oliver Lorscheid"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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e7faeac1f09c8d0f72c15c1c9ba681ab0c2a9411 | subsection | 113 | 144 | Rescaling classes | Then the map\begin{array}{cccc}
\Phi : & {\mathcal {X}}_M^f(F) & \longrightarrow & \operatorname{Hom}(k_M^f,F) \\
& [\Delta ] & \longmapsto & _{[\Delta ]}\vert _{k_M^f}
\end{array}is a bijection that is functorial in F.By Theorem REF , two Grassmann-Plücker functions \Delta and \Delta ^{\prime } that represent M are re... | {
"cite_spans": []
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"Matthew Baker",
"Oliver Lorscheid"
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