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The antidiagonal ideal \( {J}_{2143} \) for the \( 4 \times 4 \) permutation 2143 equals \( \left\langle {{x}_{11},{x}_{13}{x}_{22}{x}_{31}}\right\rangle \). | The antidiagonal complex \( {\mathcal{L}}_{2143} \) is the union of three coordinate subspaces \( {L}_{{11},{13}},{L}_{{11},{22}} \), and \( {L}_{{11},{31}} \), with ideals\n\n\[ I\left( {L}_{{11},{13}}\right) = \left\langle {{x}_{11},{x}_{13}}\right\rangle ,\;I\left( {L}_{{11},{22}}\right) = \left\langle {{x}_{11},{x}... | Yes |
Lemma 16.21 The set of complements \( {D}_{L} \) of faces \( L \in {\mathcal{L}}_{w} \) is closed under chute moves and inverse chute moves. | Proof. A pipe dream \( D \) is equal to \( {D}_{L} \) for some face \( L \in {\mathcal{L}}_{w} \) if and only if \( D \) meets every antidiagonal in \( {J}_{w} \) . Suppose that \( C \) is a chutable rectangle in \( {D}_{L} \) for \( L \in {\mathcal{L}}_{w} \) . For chutes, it is enough to show that the intersection \(... | No |
Lemma 16.22 Given a face \( L \in {\mathcal{L}}_{w} \), there is a sequence \( {L}_{0},\ldots ,{L}_{m} \) of faces of \( {\mathcal{L}}_{w} \) in which \( {L}_{0} = L \), the face \( {L}_{m} \) is top-justified and \( {L}_{e + 1} \) is obtained from \( {L}_{e} \) by either deleting \( a + \) tile or performing an invers... | Proof. Suppose that \( {D}_{L} \) for some face \( L \in {\mathcal{L}}_{w} \) is not top-justified, and has no inverse-chutable rectangles. Consider a configuration \( + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + ... | Yes |
Proposition 16.23 Let \( D \) and \( E \) be pipe dreams, with \( E \) obtained from \( D \) by a chute move. Then \( {L}_{D} \) is a facet of \( {\mathcal{L}}_{w} \) if and only if \( {L}_{E} \) is. | Proof. Suppose that \( {L}_{D} \) is not a facet. This means that deleting from \( D \) some +, let us call it \( \boxplus \), yields a pipe dream \( {D}^{\prime } \) whose subspace \( {L}_{{D}^{\prime }} \) is still a face of \( {\mathcal{L}}_{w} \) . We will show that some + may be deleted from \( E \) to yield a pip... | Yes |
Proposition 16.24 Every top-justified pipe dream is reduced, and \( \mathcal{R}\mathcal{P}\left( w\right) \) contains a unique one, called the top reduced pipe dream \( \operatorname{top}\left( w\right) \) . Every pipe dream \( D \in \mathcal{R}\mathcal{P}\left( w\right) \) can be reached by a sequence of chutes from \... | Proof. Replacing \( w \) with \( \widetilde{w} \) if necessary, it suffices to consider honest permutations, as usual. Next we show that reduced pipe dreams that are not top-justified always admit inverse chute moves. Consider a configuration \( + \) in the most eastern column containing one. To the east of this config... | Yes |
Lemma 16.25 Antidiagonals in \( {J}_{w} \smallsetminus {J}_{{\sigma }_{i}w} \) are contained in \( {\mathbf{x}}_{i \times w\left( i\right) - 1} \) and intersect row \( i \) . | Proof. If an antidiagonal in \( {J}_{w} \) is either contained in \( {\mathbf{x}}_{i - 1 \times w\left( i\right) } \) or not contained in \( {\mathbf{x}}_{i \times w\left( i\right) } \), then some rank condition causing it is in both \( r\left( w\right) \) and \( r\left( {{\sigma }_{i}w}\right) \) . Indeed, it is easy ... | No |
Corollary 16.27 The antidiagonal simplicial complex \( {\mathcal{L}}_{w} \) is pure. In the multigrading with \( \deg \left( {x}_{ij}\right) = {t}_{i} \), it has multidegree \( \mathcal{C}\left( {\mathbb{k}\left\lbrack \mathbf{x}\right\rbrack /{J}_{w};\mathbf{t}}\right) = {\mathfrak{S}}_{w}\left( \mathbf{t}\right) \) . | Proof. Purity of \( {\mathcal{L}}_{w} \) is immediate from Theorem 16.18 and the fact that all reduced pipe dreams for \( w \) have the same number of + tiles. Using purity, Theorem 8.53 and Proposition 8.49 together imply that \( \mathcal{C}\left( {\mathbb{k}\left\lbrack \mathbf{x}\right\rbrack /{J}_{w};\mathbf{t}}\ri... | Yes |
Theorem 16.28 The minors inside the Schubert determinantal ideal \( {I}_{w} \) constitute a Gröbner basis under any antidiagonal term order: | Proof. The multidegree \( \mathcal{C}\left( {\mathbb{k}\left\lbrack \mathbf{x}\right\rbrack /\operatorname{in}\left( {I}_{w}\right) ;\mathbf{t}}\right) \) equals the Schubert polynomial \( {\mathfrak{S}}_{w}\left( \mathbf{t}\right) \) by Corollary 15.44 and Corollary 8.47. As \( {J}_{w} \) is obviously contained inside... | No |
Corollary 16.29 Schubert determinantal ideals \( {I}_{w} \) are prime. | Proof. The zero set of \( {I}_{w} \) is the matrix Schubert variety \( {\bar{X}}_{w} \), which is irreducible by Theorem 15.31. Hence the radical of \( {I}_{w} \) is prime. However, Theorem 16.28 says that \( {I}_{w} \) has a squarefree initial ideal, which automatically implies that \( {I}_{w} \) is a radical ideal. | Yes |
Corollary 16.30 The double Schubert polynomial for \( w \) satisfies\n\n\[ \n{\mathfrak{S}}_{w}\left( {\mathbf{t} - \mathbf{s}}\right) = \mathop{\sum }\limits_{{D \in \mathcal{{RP}}\left( w\right) }}{\left( \mathbf{t} - \mathbf{s}\right) }^{D},\;\text{ where }\;{\left( \mathbf{t} - \mathbf{s}\right) }^{D} = \mathop{\pr... | Proof. The multidegree \( \mathcal{C}\left( {\mathbb{k}\left\lbrack \mathbf{x}\right\rbrack /{J}_{w};\mathbf{t},\mathbf{s}}\right) \) equals the double Schubert polynomial by Theorem 15.40, Corollary 8.47, and Theorem 16.28, using the fact that \( {I}_{w} = I\left( {\bar{X}}_{w}\right) \) (Corollary 16.29). Now apply a... | Yes |
Consider the Schubert determinantal ideal \( {I}_{2143} \) for the \( 4 \times 4 \) permutation 2143. This ideal has the same generators as the ideal \( {I}_{w} \) in Example 15.7, although in a bigger polynomial ring. We discussed the antidiagonal ideal \( {J}_{w} = \operatorname{in}\left( {I}_{w}\right) \) in Example... | In the multigrading where \( \deg \left( {x}_{ij}\right) = {t}_{i} - {s}_{j} \), the multidegree of \( {L}_{{i}_{1}{j}_{1},{i}_{2}{j}_{2}} \) equals \( \left( {{t}_{{i}_{1}} - {s}_{{j}_{1}}}\right) \left( {{t}_{{i}_{2}} - {s}_{{j}_{2}}}\right) \). The formula in Corollary 16.30 says that\n\n\[ \n{\mathfrak{S}}_{2143}\l... | Yes |
Lemma 16.33 The minimal number of matrices required to express a permutation matrix \( w \) as a product of simple reflections is \( l\left( w\right) \) . | Proof. For the identity matrix this is obvious, since it has length zero. Multiplying an arbitrary permutation matrix on the left by a simple reflection either increases length by 1 or decreases it by 1 ; this is a special case of Lemma 15.21. Ascending in weak order from the identity to a permutation matrix \( w \) th... | Yes |
The unique pipe dream \( {D}_{0} \) for the \( n \times n \) long permutation (antidiagonal matrix) \( {w}_{0} \) corresponds to the ordered sequence\n\n\[ Q\left( {D}_{0}\right) = \underset{Q}{{\sigma }_{n - 1}\underbrace{{\sigma }_{n - 2}{\sigma }_{n - 1}}\cdots \cdots \underbrace{{\sigma }_{2}{\sigma }_{3}\ldots {\s... | the reverse triangular reduced expression for \( {w}_{0} \) . The part of \( Q\left( {D}_{0}\right) \) arising from each row of \( {D}_{0} \) has its own underbrace. When \( n = 4 \), the above expression simplifies to \( {Q}_{0} = {\sigma }_{3}{\sigma }_{2}{\sigma }_{3}{\sigma }_{1}{\sigma }_{2}{\sigma }_{3} \) . | Yes |
The ordered sequence constructed from the pipe dream whose crossing tiles entirely fill the \( n \times n \) grid is the reverse square word | \[ {Q}_{n \times n} = \underset{\text{bottom row }}{\underbrace{{\sigma }_{n}{\sigma }_{n + 1}\ldots {\sigma }_{{2n} - 2}{\sigma }_{{2n} - 1}}}\ldots \underset{\text{second row }}{\underbrace{{\sigma }_{2}{\sigma }_{3}\ldots {\sigma }_{n}{\sigma }_{n + 1}}}\underset{\text{top row }}{\underbrace{{\sigma }_{1}{\sigma }_{... | Yes |
Lemma 16.36 Suppose that the pipe entering row \( i \) of an \( n \times n \) pipe dream \( D \) exits column \( w\left( i\right) \) for some \( n \times n \) permutation \( w \) . Multiplying the reflections in \( Q\left( D\right) \) yields the permutation matrix \( w \) . Thus \( Q\left( D\right) \) is a reduced expr... | Proof. Use induction on the number of crossing tiles: adding a + in the \( {i}^{\text{th }} \) antidiagonal at the start of the list switches the destinations of the pipes entering through rows \( i \) and \( i + 1 \) . | No |
Proposition 16.40 Antidiagonal complexes \( {\mathcal{L}}_{w} \) are subword complexes. | Proof. When \( w \) is a permutation matrix, the fact that\n\n\[ {\mathcal{L}}_{w} = \Delta \left( {{Q}_{n \times n}, w}\right) \]\n\nis a subword complex for the \( n \times n \) reverse square word is immediate from Theorem 16.18 and Lemma 16.33. When \( w \) is an arbitrary \( k \times \ell \) partial permutation, s... | No |
Proposition 16.42 Vertex-decomposable complexes are shellable. | Proof. Use induction on the number of vertices by first shelling \( {\operatorname{del}}_{\Delta }\left( v\right) \) and then shelling the cone from \( v \) over \( {\operatorname{link}}_{\Delta }\left( v\right) \) to get a shelling of \( \Delta \) . | No |
Theorem 16.43 Antidiagonal complexes are shellable and hence Cohen-Macaulay. More generally, subword complexes are vertex-decomposable. | Proof. By Proposition 16.40 and Proposition 16.42, it is enough to prove the second sentence. With \( Q = \left( {{\sigma }_{{i}_{m}},{\sigma }_{{i}_{m - 1}},\ldots ,{\sigma }_{{i}_{1}}}\right) \), it suffices by induction on the number of vertices to demonstrate that both the link and the deletion of \( {\sigma }_{{i}... | Yes |
Corollary 16.44 Schubert determinantal rings are Cohen-Macaulay. | Proof. Apply Theorems 16.43 and 8.31 to Theorem 16.28. | No |
The sequence of three matrices in Fig. 17.1 constitutes an element in the vector space \( {M}_{23} \times {M}_{34} \times {M}_{43} \) of sequences of linear maps \( {\mathbb{k}}^{2} \rightarrow {\mathbb{k}}^{3} \rightarrow {\mathbb{k}}^{4} \rightarrow {\mathbb{k}}^{3} \) (so \( {\mathbb{k}}^{r} \) consists of row vecto... | Let \( \mathbb{k}\left\lbrack \mathbf{f}\right\rbrack \) denote the coordinate ring of \( {M}_{23} \times {M}_{34} \times {M}_{43} \) . Thus \( \mathbb{k}\left\lbrack \mathbf{f}\right\rbrack \) is a polynomial ring \( 6 + {12} + {12} = {30} \) variables \( \mathbf{f} = \left\{ {f}_{\alpha \beta }^{i}\right\} \), arrang... | Yes |
Lemma 17.5 If \( \mathbf{w} \in \) Mat is a lacing diagram with precisely \( {q}_{k\ell } \) laces beginning in column \( k \) and ending in column \( \ell \), for each \( k \leq \ell \), then \( {r}_{ij}\left( \mathbf{w}\right) \) equals the number of laces passing through both column \( i \) and column \( j \) : | \[ {r}_{ij}\left( \mathbf{w}\right) = \mathop{\sum }\limits_{{k = 0}}^{i}\mathop{\sum }\limits_{{\ell = j}}^{n}{q}_{k\ell } \] | Yes |
Lemma 17.8 Every lacing diagram \( \mathbf{w} \in \) Mat is isomorphic to the direct sum of the indecomposable lacing diagrams corresponding to its laces. Two lacing diagrams are isomorphic if and only if they have the same lace array. | The (easy) proof is left to Exercise 17.2; note that the second sentence is a consequence of the first. | No |
Proposition 17.9 Every quiver representation \( \phi \in \) Mat is isomorphic to a lacing diagram \( \mathbf{w} \), whose lace array \( \mathbf{q} \) is independent of the choice of \( \mathbf{w} \) . | Proof. It suffices by Lemma 17.8 to show that \( \phi \) is isomorphic to a direct sum of indecomposables. We may as well assume that \( {r}_{0} \neq 0 \) and let \( j \) be the largest index for which the composite \( {\mathbb{k}}^{{r}_{0}} \rightarrow {\mathbb{k}}^{{r}_{j}} \) is nonzero. Choose a linearly independen... | Yes |
Consider two matrices of variables, \( {\Phi }_{1} \) and \( {\Phi }_{2} \), where \( {\Phi }_{1} \) has size \( {r}_{0} \times {r}_{1} \) and \( {\Phi }_{2} \) has size \( {r}_{1} \times {r}_{2} \). We are interested in the ideal \( I \) generated by all of the minors of size \( \rho + 1 \) in the product \( {\Phi }_{... | Suppose that \( I = {I}_{\mathbf{r}} \) for some rank array \( \mathbf{r} \). In order for the only equations generating \( I \) to be the minors in \( {\Phi }_{1}{\Phi }_{2} \), there must be no rank conditions on \( {\Phi }_{1} \) individually, and also none on \( {\Phi }_{2} \) individually. In other words, we must ... | Yes |
Example 17.14 Let \( \mathbf{r},\mathbf{q} \), and \( \mathbf{R} \) be as in Example 17.7. The Zelevinsky permutation for this data is\n\n\[ \nv\left( \mathbf{r}\right) = \left\lbrack \begin{matrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & {10} & {11} & {12} \\ 8 & 9 & 4 & 5 & {11} & 1 & 2 & 6 & {12} & 3 & 7 & {10} \end{mat... | whose permutation matrix is indicated by the \( \times \) entries in the array\n\n<table><tr><td>8</td><td></td><td>＊＊＊</td><td></td><td></td><td></td><td>＊＊＊＊</td><td></td><td>X</td><td>.</td><td>.</td><td>.</td><td>.</td></tr><tr><td>9</td><td></td><td>＊＊＊</td><td></td><td></td><td></td><td>＊＊＊＊</td><td></td><td>.</t... | Yes |
Lemma 17.16 Let \( {\Gamma \Phi } \) be the product of two matrices with entries in a commutative ring \( R \) . If \( \Gamma \) is square and \( \det \left( \Gamma \right) \) is a unit, then for each fixed \( u \in \mathbb{N} \), the ideals generated by the size \( u \) minors in \( \Phi \) and in \( {\Gamma \Phi } \)... | Proof. The result is easy when \( u = 1 \) . The case of arbitrary \( u \) reduces to the case \( u = 1 \) by noting that the minors of size \( u \) in a matrix for a map \( {R}^{k} \rightarrow {R}^{\ell } \) of free modules are simply the entries in a particular choice of matrix for the associated map \( \mathop{\bigw... | No |
Example 17.18 For a generic \( 4 \times 5 \) matrix \( {\Phi }_{1} \) and \( 5 \times 3 \) matrix \( {\Phi }_{2} \), let \( I \) be the ideal of \( 3 \times 3 \) minors in \( {\Phi }_{1}{\Phi }_{2} \) . Thus \( I = {I}_{\mathbf{r}} \) for the rank array \( \mathbf{r} \) of the lacing diagrams on the right-hand side of ... | \[ \Phi = \left\lbrack \begin{matrix} 0 & 0 & 0 & & & & \\ 0 & 0 & 0 & & & & \\ 0 & 0 & 0 & & & & \\ 0 & 0 & 0 & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & & & & & & \\ & &... | No |
Lemma 17.19 The number \( \operatorname{rank}\left( {v{\left( \mathbf{r}\right) }_{k \times \ell }}\right) \) in (17.5) is \( {r}_{ij} + \mathop{\sum }\limits_{{m = i + 1}}^{{j - 1}}{r}_{m} \) . | Proof. The coefficient on \( {q}_{\alpha \beta } \) in \( {r}_{ij} + \mathop{\sum }\limits_{{m = i + 1}}^{{j - 1}}{r}_{m} \) is the number of elements in \( \left\{ {r}_{ij}\right\} \cup \left\{ {{r}_{i + 1, i + 1},\ldots ,{r}_{j - 1, j - 1}}\right\} \) that are weakly northwest of \( {r}_{\alpha \beta } \) in the rank... | Yes |
Lemma 17.21 The multiplication map \( P \times \) Mat \( \rightarrow P \cdot \mathcal{Z}\left( \text{Mat}\right) \) that sends \( \left( {\pi ,\phi }\right) \) to the product \( \pi \mathcal{Z}\left( \phi \right) \) of matrices in \( {M}_{d} \) is an isomorphism of varieties that takes \( P \times {\Omega }_{\mathbf{r}... | Proof. It is enough to treat the case where \( {\Omega }_{\mathbf{r}} = \) Mat. Denote by \( \Phi \) the generic matrix obtained from (17.4) after replacing its blocks \( {\phi }_{i} \) by \( {\Phi }_{i} \), and let \( {\mathbf{x}}_{{v}_{ * }} \) be the block matrix of coordinate variables on \( {\bar{X}}_{{v}_{ * }} \... | Yes |
Proposition 17.22 Multiplication by \( P \) on the left preserves the matrix Schubert variety \( {\bar{X}}_{v\left( \mathbf{r}\right) } \) . In fact, \( {\bar{X}}_{v\left( \mathbf{r}\right) } \) is the closure in \( {M}_{d} \) of \( P \cdot \mathcal{Z}\left( {\Omega }_{\mathbf{r}}\right) \) . | Proof. Definition 17.12.4 and Corollary 15.33 imply that the matrix Schubert variety \( {\bar{X}}_{v\left( \mathbf{r}\right) } \) is stable under the action of \( {S}_{{r}_{0}} \times \cdots \times {S}_{{r}_{n}} \), the block diagonal permutation matrices whose blocks have sizes \( {r}_{0},\ldots ,{r}_{n} \) acting on ... | Yes |
Proposition 17.31 Let \( \mathcal{F} \) . be a \( {\mathbb{Z}}^{2d} \) -graded free resolution of \( \mathbb{k}\left\lbrack \mathbf{x}\right\rbrack /{I}_{v\left( \mathbf{r}\right) } \) over \( \mathbb{k}\left\lbrack \mathbf{x}\right\rbrack \) . If \( {\mathfrak{m}}_{\mathbf{y}} \) is the ideal of \( Y \) in \( \mathbb{... | Proof. Note that \( {\mathcal{F}}_{ \bullet }/{\mathfrak{m}}_{\mathbf{y}}{\mathcal{F}}_{ \bullet } \) is complex of \( {\mathbb{Z}}^{d} \) -graded free modules over \( \mathbb{k}\left\lbrack \mathbf{y}\right\rbrack \) . Indeed, coarsening the \( {\mathbb{Z}}^{2d} \) -grading on \( \mathbb{k}\left\lbrack \mathbf{x}\righ... | Yes |
Corollary 17.32 \( {\mathcal{K}}_{Y}\left( {\mathcal{Z}\left( {\Omega }_{\mathbf{r}}\right) ;\mathbf{t}}\right) = {\mathcal{K}}_{M}\left( {{\bar{X}}_{v\left( \mathbf{r}\right) };\mathbf{t},\overset{ \circ }{\mathbf{t}}}\right) \) . | Proof. This is immediate from Proposition 17.31, by Definition 8.32. | No |
Lemma 17.33 The \( K \) -polynomial \( {\mathcal{K}}_{\text{Mat }}\left( {{\Omega }_{\mathbf{r}};\mathbf{t}}\right) \) of \( {\Omega }_{\mathbf{r}} \) inside Mat is\n\n\[ \n{\mathcal{K}}_{\text{Mat }}\left( {{\Omega }_{\mathbf{r}};\mathbf{t}}\right) = \frac{{\mathcal{K}}_{Y}\left( {\mathcal{Z}\left( {\Omega }_{\mathbf{... | Proof. The equality \( H\left( {{\Omega }_{\mathbf{r}};\mathbf{t}}\right) = {\mathcal{K}}_{\text{Mat }}\left( {{\Omega }_{\mathbf{r}};\mathbf{t}}\right) H\left( {\text{Mat; }\mathbf{t}}\right) \) of Hilbert series (which are well-defined by positivity of the grading of \( \mathbb{k}\left\lbrack \mathbf{f}\right\rbrack ... | Yes |
Theorem 17.34 The ordinary quiver polynomial \( {\mathcal{Q}}_{\mathbf{r}}\left( {\mathbf{t} - \overset{ \circ }{\mathbf{t}}}\right) \) is the \( \overset{ \circ }{\mathbf{s}} = \overset{ \circ }{\mathbf{t}} \) specialization of the double quiver polynomial \( {\mathcal{Q}}_{\mathbf{r}}\left( {\mathbf{t} - \overset{ \c... | Proof. After clearing denominators in Lemma 17.33, substitute using Corollary 17.32 to get\n\n\[ \n{\mathcal{K}}_{\text{Mat }}\left( {{\Omega }_{\mathbf{r}};\mathbf{t}}\right) {\mathcal{K}}_{M}\left( {{\bar{X}}_{{v}_{ * }};\mathbf{t},\overset{ \circ }{\mathbf{t}}}\right) = {\mathcal{K}}_{M}\left( {{\bar{X}}_{v\left( \m... | Yes |
Theorem 17.36 The double quiver polynomial for ranks \( \mathbf{r} \) equals the sum\n\n\[ \n{\mathcal{Q}}_{\mathbf{r}}\left( {\mathbf{t} - \overset{ \circ }{\mathbf{s}}}\right) = \mathop{\sum }\limits_{{D \in \mathcal{R}\mathcal{P}\left( {v\left( \mathbf{r}\right) }\right) }}{\left( \mathbf{t} - \overset{ \circ }{\mat... | Proof. This follows from Definition 17.30 and Corollary 16.30, using the fact that every pipe dream \( D \in \mathcal{R}\mathcal{P}\left( {v\left( \mathbf{r}\right) }\right) \) contains the subdiagram \( D\left( {v}_{ * }\right) \) and that \( \mathcal{R}\mathcal{P}\left( {v}_{ * }\right) \) consists of the single pipe... | Yes |
The partial permutations arising from the pipe dream in Example 17.35 come from the following partial reduced pipe dreams: | These send each number along the top either to the number along the bottom connected to it by a pipe (if such a pipe exists), or to nowhere. It is easy to see the pictorial lacing diagram \( \mathbf{w}\left( D\right) \) from these pictures. Indeed, removing all segments of all pipes not contributing to one of the parti... | No |
Proposition 17.40 Every reduced pipe dream \( D \in \mathcal{R}\mathcal{P}\left( {v\left( \mathbf{r}\right) }\right) \) gives rise to a lacing diagram \( \mathbf{w}\left( D\right) \) representing a partial permutation list with ranks \( \mathbf{r} \) . | Proof. Each \( \times \) entry in the permutation matrix for \( v\left( \mathbf{r}\right) \) corresponds to a pipe in \( D \) entering due north of it and exiting due west of it. The permutation \( v\left( \mathbf{r}\right) \) was specifically constructed to have exactly \( {q}_{ij} \) entries \( \times \) (for \( i \l... | No |
Theorem 17.41 The quiver polynomial \( {\mathcal{Q}}_{\mathbf{r}}\left( {\mathbf{t} - \overset{ \circ }{\mathbf{t}}}\right) \) equals the sum\n\n\[ \n{\mathcal{Q}}_{\mathbf{r}}\left( {\mathbf{t} - \mathring{\mathbf{t}}}\right) = \mathop{\sum }\limits_{{\mathbf{w} \in W\left( \mathbf{r}\right) }}{\mathfrak{S}}_{{w}_{1}}... | This statement was discovered by Knutson, Miller, and Shimozono, who at first proved only that the expansion on the right-hand side has positive coefficients. After publicizing their weaker statement and conjecturing the above precise statement, independent (and quite different) proofs of the conjecture were given by t... | No |
Consider the partition \( 2 + 1 + 1 \) of \( n = 4 \) . The ideal \( {I}_{2 + 1 + 1} \) equals \( \left\langle {{x}^{2},{xy},{y}^{3}}\right\rangle \) . | The monomial \( {x}^{2} \) would be the first box after the bottom row, whereas \( {xy} \) would nestle in the nook of the \( \mathrm{L} \), and \( {y}^{3} \) would lie atop the first column. \( \diamond \) | No |
Lemma 18.2 Given any colength \( n \) ideal \( I \), the image of \( {V}_{m} \) spans the quotient \( \mathbb{C}\left\lbrack {x, y}\right\rbrack /I \) as a vector space whenever \( m \geq n \) . | Proof. The \( n \) monomials outside any initial monomial ideal of \( I \) span the quotient \( \mathbb{C}\left\lbrack {x, y}\right\rbrack /I \), and these monomials must lie inside \( {V}_{m} \) . | Yes |
Theorem 18.4 The affine varieties \( {U}_{\lambda } \) form an open cover of the subset \( {H}_{n} \subset {\operatorname{Gr}}^{n}\left( {V}_{m}\right) \) for \( m \geq n + 1 \), thereby endowing \( {H}_{n} \) with the structure of a quasiprojective variety (i.e., an open subvariety of a projective variety). | Proof. The sets \( {U}_{\lambda } \) cover \( {H}_{n} \) by Lemma 18.2, and each set \( {U}_{\lambda } \) is locally closed in \( {\operatorname{Gr}}^{n}\left( {V}_{m}\right) \) by the above discussion. (We will explain near the beginning of Section 18.2 why we assumed \( m \geq n + 1 \) instead of \( m \geq n \) .) \(... | Yes |
Example 18.5 Take \( n = 4 \) and \( \lambda \) the partition \( 2 + 1 + 1 \) of Example 18.1. Every ideal \( I \) in \( {U}_{2 + 1 + 1} \) is generated by three of the polynomials in (18.2): | \[ \left\langle {{\underline{x}}^{2} - a{y}^{2} - {bx} - {py} - q,\underline{xy} - c{y}^{2} - {dy} - {ex} - r,\underline{{y}^{3}} - f{y}^{2} - {gy} - {hx} - s}\right\rangle . \] Here, we abbreviate \( a = {c}_{02}^{20}, p = {c}_{01}^{20} \), and so on. This ideal lies in \( {U}_{2 + 1 + 1} \) if and only if its three g... | Yes |
Theorem 18.7 The Hilbert scheme \( {H}_{n} \) is a smooth and irreducible complex algebraic variety of dimension \( {2n} \) . | The variety structure in Theorem 18.7 is the same as the one from Theorem 18.4, although it is not obvious from the latter that this structure is independent of \( m \) . This important fact can be deduced using the smoothness of \( {H}_{n} \) along with the fact that projection \( {V}_{m + 1} \rightarrow {V}_{m} \) ma... | No |
Lemma 18.8 Every point \( I \in {H}_{n} \) is connected to a monomial ideal by a rational curve. | Proof. Choosing a term order and taking a Gröbner basis of \( I \) yields a family of ideals parametrized by the coordinate variable \( t \) on the affine line. Such a Gröbner degeneration is a flat family \( {I}_{t} \) over the affine line by Proposition 8.26. When \( t = 1 \) we get \( I \) back, and when \( t = 0 \)... | Yes |
Consider the ideal \( I = \left\langle {{x}^{2} - {xy},{y}^{2} - {xy},{x}^{2}y, x{y}^{2}}\right\rangle \), which lies in the chart \( {U}_{2 + 2} \) discussed in Example 18.6. Now replace \( y \) by \( {ty} \) in every polynomial \( f \in I \), and observe what happens as \( t \) goes to 0 . Finding polynomials in \( I... | Our rational curve in \( {H}_{4} \) is given by\n\n\[ \n{I}_{t} = \left\langle {{x}^{2} - {txy},{xy} - {t}^{2}{y}^{2},{x}^{2}y, x{y}^{2},{y}^{3}}\right\rangle .\n\]\n\nThis represents a flat family because the quotient ring \( \mathbb{C}\left\lbrack {x, y}\right\rbrack \left\lbrack t\right\rbrack /{I}_{t} \) is a free ... | Yes |
For every partition \( \lambda \) of \( n \), the point \( {I}_{\lambda } \in {H}_{n} \) lies in the closure of the locus \( {\left( {S}^{n}{\mathbb{C}}^{2}\right) }^{ \circ } \) of all radical ideals in the Hilbert scheme \( {H}_{n} \) . | Consider the set of exponent vectors \( \left( {h, k}\right) \) on monomials \( {x}^{h}{y}^{k} \) outside \( {I}_{\lambda } \) . This set constitutes a collection of \( n \) points in \( {\mathbb{N}}^{2} \subset {\mathbb{C}}^{2} \) . The radical ideal of these points is denoted by \( {I}_{\lambda }^{\prime } \) and cal... | Yes |
Example 18.11 The distraction of \( {I}_{2 + 1 + 1} = \left\langle {{x}^{2},{xy},{y}^{3}}\right\rangle \) is the ideal | \[ {I}_{2 + 1 + 1}^{\prime } = \langle x\left( {x - 1}\right) ,{xy}, y\left( {y - 1}\right) \left( {y - 2}\right) \rangle . \] | Yes |
Proposition 18.12 The Hilbert scheme \( {H}_{n} \) is connected. | Proof. We connect any two points \( I \) and \( J \) in \( {H}_{n} \) by a path as follows. Go from \( I \) to any initial monomial ideal \( {I}_{\lambda } \) and then to its distraction \( {I}_{\lambda }^{\prime } \) . Go from \( J \) to any initial monomial ideal \( {I}_{\nu } \) and then to its distraction \( {I}_{\... | Yes |
Proposition 18.14 For each partition \( \lambda \) of \( n \), the local ring \( {\left( {H}_{n}\right) }_{{I}_{\lambda }} \) of the Hilbert scheme \( {H}_{n} \) at \( {I}_{\lambda } \) has embedding dimension at most \( {2n} \) ; that is, the maximal ideal \( {\mathfrak{m}}_{{I}_{\lambda }} \) satisfies \( {\dim }_{\m... | Proof. Identify each variable \( {c}_{hk}^{rs} \) with an arrow pointing from the box \( {hk} \in \lambda \) to the box \( {rs} \notin \lambda \) (see Example 18.16). Allow arrows starting in boxes with \( h < 0 \) or \( k < 0 \), but set them equal to zero. The arrows lie inside and in fact generate the maximal ideal ... | Yes |
Example 18.20 For \( n = 3 \), the ideal \( {I}_{\text{diag }} \) has five minimal generators \( {\Delta }_{D} \) : | \[ {I}_{\text{diag }} = \left\langle {{x}_{1} - {x}_{2},{y}_{1} - {y}_{2}}\right\rangle \cap \left\langle {{x}_{1} - {x}_{3},{y}_{1} - {y}_{3}}\right\rangle \cap \left\langle {{x}_{2} - {x}_{3},{y}_{2} - {y}_{3}}\right\rangle \] \[ = \left\langle {\det \left\lbrack \begin{matrix} 1 & 1 & 1 \\ {x}_{1} & {x}_{2} & {x}_{3... | Yes |
Theorem 18.21 (Haiman’s \( n \) ! Theorem and \( {\left( n + 1\right) }^{n - 1} \) Theorem)\n\n1. If \( \lambda \) is a partition of \( n \), then the set of all polynomials obtained from \( {\Delta }_{\lambda } \) by applying linear partial differential operators with constant coefficients span a vector space of dimen... | Part 1 of Theorem 18.21 can be reformulated in ideal-theoretic terms as follows. A linear partial differential operator with constant coefficients is by definition a polynomial\n\n\[ p\left( {\partial \mathbf{x},\partial \mathbf{y}}\right) = p\left( {\frac{\partial }{\partial {x}_{1}},\ldots ,\frac{\partial }{\partial ... | Yes |
Let \( n = 3 \) and \( \lambda = 2 + 1 \) . Then\n\n\[ \n{\Delta }_{\lambda } = {x}_{1}{y}_{2} - {x}_{2}{y}_{1} + {x}_{3}{y}_{1} - {x}_{1}{y}_{3} - {x}_{3}{y}_{2} + {x}_{2}{y}_{3} \n\] | By differentiating \( {\Delta }_{\lambda } \), we first get the differences \( {x}_{i} - {x}_{j} \) and \( {y}_{i} - {y}_{j} \), and next the constants. Together they span a vector space of dimension \( 3! = 6 \) . The annihilating ideal of \( {\Delta }_{\lambda } \) is\n\n\[ \n{J}_{\lambda } = \left\langle {{x}_{1} + ... | Yes |
Example 18.23 Let \( n = 3 \) and \( \lambda = 2 + 1 \) . The ideal \( {L}_{\lambda } \) is generated by\n\n\[ \n{x}_{1}^{2} - a{x}_{1} - b{y}_{1} - c,\;{x}_{1}{y}_{1} - d{x}_{1} - e{y}_{1} - f,\;{y}_{1}^{2} - g{x}_{1} - h{y}_{1} - i, \]\n\n\[ \n\begin{array}{lll} {x}_{2}^{2} - a{x}_{2} - b{y}_{2} - c, & {x}_{2}{y}_{2}... | \[ \n{y}_{1} + {y}_{2} + {y}_{3} - d - h,{x}_{1} + {x}_{2} + {x}_{3} - a - e, \]\n\n\[ \n{y}_{3}^{2} - {x}_{3}g - {y}_{3}h - i,\;{y}_{2}{y}_{3} - {y}_{2}d - {y}_{3}d + {x}_{2}g + {x}_{3}g - {eg} + {dh} + i,\;{y}_{2}^{2} - {x}_{2}g - {y}_{2}h - i, \]\n\n\[ \n{x}_{3}^{2} - {x}_{3}a - {y}_{3}b - c,{x}_{2}{x}_{3} + {y}_{2}... | Yes |
Theorem 18.24 (Haiman) The isospectral Hilbert scheme is Gorenstein. | Consider now the morphism \( {X}_{n} \rightarrow {H}_{n} \) in (18.6). The base is smooth by Theorem 18.7. The generic fiber is reduced of length \( n \) !. It is given by all permutations of \( n \) distinct points \( \left( {{x}_{i},{y}_{i}}\right) \) in \( {\mathbb{C}}^{2} \) . Theorem 18.24 implies that all special... | No |
Lemma 18.25 The fiber of the morphism \( {X}_{n} \rightarrow {H}_{n} \) over the torus-fixed point \( {I}_{\lambda } \in {H}_{n} \) is the zero-dimensional scheme defined by the ideal \( {J}_{\lambda } \) (18.5). | For \( n = 3 \) and \( \lambda = 2 + 1 \), this lemma was confirmed computationally in Example 18.23. | No |
Theorem 18.29 The tangent space to the Hilbert scheme \( {H}_{n}^{d} \) at any point \( I \in {H}_{n}^{d} \) is isomorphic as \( \mathbb{C} \) -vector space to the module \( {\operatorname{Hom}}_{\mathbb{C}\left\lbrack \mathbf{x}\right\rbrack }\left( {I,\mathbb{C}\left\lbrack \mathbf{x}\right\rbrack /I}\right) \) . | This theorem is derived from the universal property of the Hilbert scheme, a topic we will only briefly mention in Section 18.5. If \( I = {I}_{\lambda } \) is a monomial ideal, then the image of the parameter \( {c}_{\mathbf{v}}^{\mathbf{u}} \) in the tangent space corresponds to the unique \( \mathbb{C} \) -linear ma... | No |
The Hilbert scheme \( {H}_{n}^{d} \) is not smooth if \( n > d \geq 3 \) . In fact, the square of the maximal ideal in \( \mathbb{C}\left\lbrack \mathbf{x}\right\rbrack \) is a singular point of \( {H}_{d + 1}^{d} \) . | Proof. As before, the Hilbert scheme \( {H}_{n}^{d} \) contains the locus \( {\left( {S}^{n}{\mathbb{C}}^{d}\right) }^{ \circ } \) of radical ideals as an open subvariety. This subvariety is smooth of dimension \( {dn} \) . It parametrizes unordered configurations of \( n \) distinct points in \( {\mathbb{C}}^{d} \), o... | Yes |
Consider the Hilbert scheme \( {H}_{4}^{3} \) of four points in affine 3-space. One of the monomial ideals in \( {H}_{4}^{3} \) is the square\n\n\[ \n{I}_{\lambda } = \langle x, y, z{\rangle }^{2} = \left\langle {{x}^{2},{xy},{xz},{y}^{2},{yz},{z}^{2}}\right\rangle \n\]\n\nof the maximal ideal \( \langle x, y, z\rangle... | For instance,\n\n\[ \n{d}_{2} = {c}_{3}{c}_{10} + {c}_{2}{c}_{10} - {c}_{4}{c}_{6} - {c}_{4}{c}_{5} \n\]\n\nThe remaining equations in the 18 parameters \( {c}_{j} \) are all quadratic. They generate a prime ideal of dimension 12. Hence the Hilbert scheme \( {H}_{4}^{3} \) is irreducible of dimension 12, but its tangen... | Yes |
Theorem 18.32 (Iarrobino) If \( d \geq 3 \) and \( n \gg d \) then the Hilbert scheme \( {H}_{n}^{d} \) has more than one irreducible component and its dimension exceeds dn. | Proof. The radical locus \( {\left( {S}^{n}{\mathbb{C}}^{d}\right) }^{ \circ } \) is an open subvariety of \( {H}_{n}^{d} \) . Let \( {R}_{n}^{d} \) denote its closure in \( {H}_{n}^{d} \) . Since \( {\left( {S}^{n}{\mathbb{C}}^{d}\right) }^{ \circ } \) is smooth and irreducible of dimension \( {dn} \), we know that \(... | Yes |
For \( d = 3 \), the smallest value of \( n \) for which (18.12) exceeds the dimension \( {3n} \) of the radical locus \( {\left( {S}^{n}{\mathbb{C}}^{3}\right) }^{ \circ } \) is \( n = {102} \) . | For that value, we have \( r = 7 \) and the lower and upper bounds in (18.11) are 84 and 120. Hence (18.12) is \( {18}^{2} = {324} \) while \( {3n} = {306} \) . In concrete terms: there exist ideals \( {J}_{W} \) of colength 102 in \( \mathbb{C}\left\lbrack {x, y, z}\right\rbrack \) that are not in the closure of the l... | Yes |
Example 18.36 Consider the case \( d = 3 \) and \( n = 8 \) . There are 160 monomial ideals of colength 8 in \( \mathbb{C}\left\lbrack {x, y, z}\right\rbrack \) . These 160 ideals come in 33 types modulo permutations of the three variables. The ideals with the most generators are | \[ \left\langle {{xy},{xz}, y{z}^{2},{y}^{2}z,{x}^{2},{y}^{3},{z}^{4}}\right\rangle \text{ and }\left\langle {{xy}, x{z}^{2}, y{z}^{2},{y}^{2}z,{x}^{2},{z}^{3},{y}^{3}}\right\rangle . \] The tangent space of the Hilbert scheme \( {H}_{8}^{3} \) at these singular points has dimension 32 in both cases. On the other hand,... | Yes |
(i) If \( A = \{ 0\} \) is the one-element group, then \( {\mathcal{H}}_{\mathbb{C}\left\lbrack {x, y}\right\rbrack }^{h} \) is the Hilbert scheme of \( h\left( 0\right) \) points in the affine plane \( {\mathbb{C}}^{2} \) . | We saw in Theorem 18.7 that this Hilbert scheme is smooth and irreducible of dimension \( {2n} \). | No |
Example 18.48 Let \( n = 3 \), fix the \( {\mathbb{Z}}^{2} \) -grading in Example 18.41, and let \( h = {h}_{M} \) be the Hilbert function with Hilbert series (18.15). The multi-graded Hilbert scheme \( {H}_{\mathbb{C}\left\lbrack {x, y, z}\right\rbrack }^{h} \) is the reduced union of two projective lines \( {\mathbb{... | \[ \left\langle {{x}^{3}, x{y}^{2},{x}^{2}y,{y}^{3},{a}_{0}{x}^{2}z - {a}_{1}{xy},{b}_{0}{xyz} - {b}_{1}{y}^{2},{y}^{2}z,{z}^{2}}\right\rangle \;\text{ with }\;{a}_{1}{b}_{1} = 0. \] Here, \( \left( {{a}_{0} : {a}_{1}}\right) \) and \( \left( {{b}_{0} : {b}_{1}}\right) \) are coordinates on two projective lines. This H... | Yes |
Let \( A = \mathbb{Z} \) and give \( \mathbb{C}\left\lbrack \mathbf{x}\right\rbrack \) the standard grading with \( \deg \left( {x}_{i}\right) = 1 \) for \( i = 1,\ldots, n \) . Consider the following family of Hilbert functions \( h \) . Let \( p\left( t\right) \) be any univariate polynomial with \( p\left( \mathbb{N... | The multigraded Hilbert scheme \( {H}_{\mathbb{C}\left\lbrack \mathbf{x}\right\rbrack }^{h} \) parametrizes all subschemes of projective space \( {\mathbb{P}}^{n - 1} \) with Hilbert polynomial \( p \) . This is the classical Hilbert scheme due to Grothendieck. It is known to be connected [Har66a]. | No |
Proposition 1.2.5. The Dedekind eta function satisfies the transformation law\n\n\\[ \eta \\left( {-1/\\tau }\\right) = \\sqrt{-{i\\tau }}\\eta \\left( \\tau \\right) ,\\;\\tau \\in \\mathcal{H}. \\] | Proof. Compute the logarithmic derivative\n\n\\[ \\frac{d}{d\\tau }\\log \\left( {\\eta \\left( \\tau \\right) }\\right) = \\frac{\\pi i}{12} - {2\\pi i}\\mathop{\\sum }\\limits_{{d = 1}}^{\\infty }\\frac{d{q}^{d}}{1 - {q}^{d}} = \\frac{\\pi i}{12} - {2\\pi i}\\mathop{\\sum }\\limits_{{d = 1}}^{\\infty }d\\mathop{\\sum... | Yes |
Lemma 1.3.1. Consider two lattices \( \Lambda = {\omega }_{1}\mathbb{Z} \oplus {\omega }_{2}\mathbb{Z} \) and \( {\Lambda }^{\prime } = {\omega }_{1}^{\prime }\mathbb{Z} \oplus {\omega }_{2}^{\prime }\mathbb{Z} \) with \( {\omega }_{1}/{\omega }_{2} \in \mathcal{H} \) and \( {\omega }_{1}^{\prime }/{\omega }_{2}^{\prim... | Proof. Exercise 1.3.1. | No |
Proposition 1.3.2. Suppose \( \varphi : \mathbb{C}/\Lambda \rightarrow \mathbb{C}/{\Lambda }^{\prime } \) is a holomorphic map between complex tori. Then there exist complex numbers \( m, b \) with \( {mA} \subset {A}^{\prime } \) such that \( \varphi \left( {z + \Lambda }\right) = {mz} + b + {\Lambda }^{\prime } \) . ... | Proof. (Sketch.) The key is to lift \( \varphi \) to a holomorphic map \( \widetilde{\varphi } : \mathbb{C} \rightarrow \mathbb{C} \) by using topology. (The plane is the so-called universal covering space of the torus - see a topology text such as [Mun00] for the definition and the relevant lifting theorem.) With the ... | Yes |
Corollary 1.3.3. Suppose \( \varphi : \mathbb{C}/\Lambda \rightarrow \mathbb{C}/{\Lambda }^{\prime } \) is a holomorphic map between complex tori, \( \varphi \left( {z + \Lambda }\right) = {mz} + b + {\Lambda }^{\prime } \) with \( {m\Lambda } \subset {\Lambda }^{\prime } \) . Then the following are equivalent:\n\n(1) ... | Proof. Exercise 1.3.2. | No |
Proposition 1.4.1. Let \( \wp \) be the Weierstrass function with respect to a lattice A. Then\n\n(a) The Laurent expansion of \( \wp \) is\n\n\[ \wp \left( z\right) = \frac{1}{{z}^{2}} + \mathop{\sum }\limits_{\substack{{n = 2} \\ {n\text{ even }} }}^{\infty }\left( {n + 1}\right) {G}_{n + 2}\left( \Lambda \right) {z}... | Proof. (Sketch.) For (a), if \( \left| z\right| < \left| \omega \right| \) then\n\n\[ \frac{1}{{\left( z - \omega \right) }^{2}} - \frac{1}{{\omega }^{2}} = \frac{1}{{\omega }^{2}}\left( {\frac{1}{{\left( 1 - z/\omega \right) }^{2}} - 1}\right) \]\n\nand the geometric series squares to \( \mathop{\sum }\limits_{{n = 0}... | No |
Corollary 1.4.2. The function \( \Delta \) is nonvanishing on \( \mathcal{H} \) . That is, \( \Delta \left( \tau \right) \neq 0 \) for all \( \tau \in \mathcal{H} \) . | Proof. For any \( \tau \in \mathcal{H} \), specialize the lattice \( \Lambda \) in the proposition to \( {\Lambda }_{\tau } \) . By part (c) of the proposition, the cubic polynomial \( {p}_{\tau }\left( x\right) = 4{x}^{3} - {g}_{2}\left( \tau \right) x - {g}_{3}\left( \tau \right) \) has distinct roots. Exercise 1.4.4... | No |
Proposition 1.4.3. Given an elliptic curve (1.8), there exists a lattice \( \Lambda \) such that \( {a}_{2} = {g}_{2}\left( \Lambda \right) \) and \( {a}_{3} = {g}_{3}\left( \Lambda \right) \) . | Proof. The case \( {a}_{2} = 0 \) and the case \( {a}_{3} = 0 \) are Exercise 1.4.5. For the case \( {a}_{2} \neq 0 \) and \( {a}_{3} \neq 0 \), since \( j : \mathcal{H} \rightarrow \mathbb{C} \) surjects there exists \( \tau \in \mathcal{H} \) such that \( j\left( \tau \right) = {1728}{a}_{2}^{3}/\left( {{a}_{2}^{3} -... | No |
Corollary 2.1.2. For any congruence subgroup \( \Gamma \) of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \), the modular curve \( Y\left( \Gamma \right) \) is Hausdorff. | Proof. Let \( \pi \left( {\tau }_{1}\right) \) and \( \pi \left( {\tau }_{2}\right) \) be distinct points in \( Y\left( \Gamma \right) \) . Take neighborhoods \( {U}_{1} \) of \( {\tau }_{1} \) and \( {U}_{2} \) of \( {\tau }_{2} \) as in Proposition 2.1.1. Since \( \gamma \left( {\tau }_{1}\right) \neq {\tau }_{2} \) ... | Yes |
Corollary 2.2.3. Let \( \Gamma \) be a congruence subgroup of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) . Each point \( \tau \in \) \( \mathcal{H} \) has a neighborhood \( U \) in \( \mathcal{H} \) such that \[ \text{for all}\gamma \in \Gamma \text{, if}\gamma \left( U\right) \cap U \neq \varnothing \text{then}... | Now given any point \( \pi \left( \tau \right) \in Y\left( \Gamma \right) \), take a neighborhood \( U \) as in the corollary. Define \( \psi : U \rightarrow \mathbb{C} \) to be \( \psi = \rho \circ \delta \) where \( \delta = {\delta }_{\tau } \) and \( \rho \) is the power function \( \rho \left( z\right) = {z}^{h} \... | No |
Lemma 2.3.1. The map \( \pi : \mathcal{D} \rightarrow Y\left( 1\right) \) surjects, where \( \pi \) is the natural projection \( \pi \left( \tau \right) = {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \tau \) . | Proof. Given \( \tau \in \mathcal{H} \) it suffices to show that \( \tau \) is \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) -equivalent to some point in \( \mathcal{D} \) . Repeatedly apply one of \( \left\lbrack \begin{matrix} 1 & \pm 1 \\ 0 & 1 \end{matrix}\right\rbrack : \tau \mapsto \tau \pm 1 \) to translate ... | Yes |
Corollary 2.3.4. The elliptic points for \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) are \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) i \) and \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) {\mu }_{3} \) where \( {\mu }_{3} = {e}^{{2\pi i}/3} \) . The modular curve \( Y\left( 1\right) = {\mathrm{{SL}}}_{2}\left... | Proof. The fixed points in \( \mathcal{H} \) of the matrices in Proposition 2.3.3 are \( i \) and \( {\mu }_{3} \) . The first statement follows (Exercise 2.3.5(a)). The second statement follows since \( i \) and \( {\mu }_{3} \) are not equivalent under \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) . The third sta... | No |
Corollary 2.3.5. Let \( \Gamma \) be a congruence subgroup of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \). The modular curve \( Y\left( \Gamma \right) \) has finitely many elliptic points. For each elliptic point \( \tau \) of \( \Gamma \) the isotropy subgroup \( {\Gamma }_{\tau } \) is finite cyclic. | Proof. If \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) = \mathop{\bigcup }\limits_{{j = 1}}^{d}\Gamma {\gamma }_{j} \) then the elliptic points of \( Y\left( \Gamma \right) \) are a subset of \( {E}_{\Gamma } = \left\{ {\Gamma {\gamma }_{j}\left( i\right) ,\Gamma {\gamma }_{j}\left( {\mu }_{3}\right) : 1 \leq j \leq ... | Yes |
Lemma 2.4.1. The modular curve \( X\left( 1\right) = {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \smallsetminus {\mathcal{H}}^{ * } \) has one cusp. For any congruence subgroup \( \Gamma \) of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) the modular curve \( X\left( \Gamma \right) \) has finitely many cusps. | ## Proof. Exercise 2.4.1.\n\nThe topology on \( {\mathcal{H}}^{ * } \) consisting of its intersections with open complex disks (including disks \( \{ z : \left| z\right| > r\} \cup \{ \infty \} \) ) contains too many points of \( \mathbb{Q} \cup \{ \infty \} \) in each neighborhood to make the quotient \( X\left( \Gamm... | No |
Theorem 3.1.1. Let \( \Gamma \) be a congruence subgroup of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \). Let \( f : X\left( \Gamma \right) \rightarrow \) \( X\left( 1\right) \) be natural projection, and let \( d \) denote its degree. Let \( {\varepsilon }_{2} \) and \( {\varepsilon }_{3} \) denote the number of ... | Proof. Exercise 3.1.3(b). | No |
Corollary 3.1.2. Let \( \Gamma, g,{\varepsilon }_{2},{\varepsilon }_{3} \), and \( {\varepsilon }_{\infty } \) be as above. Then\n\n\[ \n{2g} - 2 + \frac{{\varepsilon }_{2}}{2} + \frac{2{\varepsilon }_{3}}{3} + {\varepsilon }_{\infty } > 0.\n\] | Proof. Exercise 3.1.3(c). | No |
Theorem 3.3.1. Let \( k \in \mathbb{N} \) be even and let \( \Gamma \) be a congruence subgroup of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) . The map \[ \omega : {\mathcal{A}}_{k}\left( \Gamma \right) \rightarrow {\Omega }^{\otimes k/2}\left( {X\left( \Gamma \right) }\right) \] \[ f \mapsto {\left( {\omega }_{... | Proof. The map \( \omega \) is defined since we have just constructed \( \omega \left( f\right) \) . Clearly \( \omega \) is \( \mathbb{C} \) -linear and injective. And \( \omega \) is surjective because every \( \left( {\omega }_{j}\right) \in {\Omega }^{\otimes k/2}\left( {X\left( \Gamma \right) }\right) \) pulls bac... | Yes |
Theorem 3.4.1 (Riemann-Roch). Let \( X \) be a compact Riemann surface of genus \( g \) . Let \( \operatorname{div}\left( \lambda \right) \) be a canonical divisor on \( X \) . Then for any divisor \( D \in \operatorname{Div}\left( X\right) \) | \[ \ell \left( D\right) = \deg \left( D\right) - g + 1 + \ell \left( {\operatorname{div}\left( \lambda \right) - D}\right) . \] | Yes |
Corollary 3.4.2. Let \( X, g,\operatorname{div}\left( \lambda \right) \), and \( D \) be as above. Then\n\n(a) \( \ell \left( {\operatorname{div}\left( \lambda \right) }\right) = g \) .\n\n(b) \( \deg \left( {\operatorname{div}\left( \lambda \right) }\right) = {2g} - 2 \) .\n\n(c) If \( \deg \left( D\right) < 0 \) then... | Proof. Note \( \ell \left( 0\right) = 1 \) since \( \operatorname{div}\left( f\right) \geq 0 \) only for the constant functions in \( \mathbb{C}{\left( X\right) }^{ * } \), and so letting \( D = 0 \) in the Riemann-Roch Theorem gives \( 1 = - g + 1 + \ell \left( {\operatorname{div}\left( \lambda \right) }\right) \), pr... | Yes |
Theorem 3.5.2. The modular forms of weight 0 are \( {\mathcal{M}}_{0}\left( {{\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) }\right) = \mathbb{C} \) . For any nonzero even integer \( k < 4,{\mathcal{M}}_{k}\left( {{\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) }\right) = \{ 0\} \) . For any even integer \( k < 4,{\mathcal{S}... | The proof is Exercise 3.5.3. | No |
Theorem 3.6.1. Let \( k \) be an odd integer. Let \( \Gamma \) be a congruence subgroup of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) . If \( \Gamma \) contains the negative identity matrix \( - I \) then \( {\mathcal{M}}_{k}\left( \Gamma \right) = \) \( {\mathcal{S}}_{k}\left( \Gamma \right) = \{ 0\} \) . If \(... | \[ \dim \left( {{\mathcal{M}}_{k}\left( \Gamma \right) }\right) = \left\{ \begin{array}{ll} \left( {k - 1}\right) \left( {g - 1}\right) + \left\lfloor \frac{k}{3}\right\rfloor {\varepsilon }_{3} + \frac{k}{2}{\varepsilon }_{\infty }^{\text{reg }} + \frac{k - 1}{2}{\varepsilon }_{\infty }^{\text{irr }} & \text{ if }k \g... | Yes |
Corollary 3.7.2. The number of elliptic points for \( {\Gamma }_{0}\left( N\right) \) is\n\n\[ \n{\varepsilon }_{2}\left( {{\Gamma }_{0}\left( N\right) }\right) = \left\{ \begin{array}{ll} \mathop{\prod }\limits_{{p \mid N}}\left( {1 + \left( \frac{-1}{p}\right) }\right) & \text{ if }4 \nmid N, \\ 0 & \text{ if }4 \mid... | Proof. This is an application of beginning algebraic number theory; see for example Chapter 9 of [IR92] for the results to quote. For period 3, the ring \( A = \mathbb{Z}\left\lbrack {\mu }_{6}\right\rbrack \) is a principal ideal domain and its maximal ideals are\n\n- for each prime \( p \equiv 1\left( {\;\operatornam... | No |
Lemma 3.8.1. Let \( s = a/c \) and \( {s}^{\prime } = {a}^{\prime }/{c}^{\prime } \) be elements of \( \mathbb{Q} \cup \{ \infty \} \), with \( \gcd \left( {a, c}\right) = \gcd \left( {{a}^{\prime },{c}^{\prime }}\right) = 1 \) . Then for any \( \gamma \in {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \), \[ {s}^{\prime ... | Proof. Assume that \( c \neq 0 \) and \( {c}^{\prime } \neq 0 \) . Then for \( \gamma = \left\lbrack \begin{array}{ll} p & q \\ r & t \end{array}\right\rbrack \in {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \), \[ {s}^{\prime } = \gamma \left( s\right) \Leftrightarrow \frac{{a}^{\prime }}{{c}^{\prime }} = \frac{{pa} + ... | No |
Lemma 3.8.2. Let \( \left\lbrack \begin{array}{l} a \\ c \end{array}\right\rbrack \) and \( \left\lbrack \begin{array}{l} {a}^{\prime } \\ {c}^{\prime } \end{array}\right\rbrack \) be as above. Then \[ \left\lbrack \begin{array}{l} {a}^{\prime } \\ {c}^{\prime } \end{array}\right\rbrack = \gamma \left\lbrack \begin{arr... | Proof. \ | No |
Proposition 3.8.3. Let \( s = a/c \) and \( {s}^{\prime } = {a}^{\prime }/{c}^{\prime } \) be elements of \( \mathbb{Q} \cup \{ \infty \} \) with \( \gcd \left( {a, c}\right) = \gcd \left( {{a}^{\prime },{c}^{\prime }}\right) = 1 \) . Then\n\n\[ \Gamma \left( N\right) {s}^{\prime } = \Gamma \left( N\right) s \Leftright... | Proof. For the first equivalence,\n\n\[ \Gamma \left( N\right) {s}^{\prime } = \Gamma \left( N\right) s \Leftrightarrow {s}^{\prime } = \gamma \left( s\right) \text{ for some }\gamma \in \Gamma \left( N\right) \]\n\n\[ \Leftrightarrow \left\lbrack \begin{array}{l} {a}^{\prime } \\ {c}^{\prime } \end{array}\right\rbrack... | Yes |
Lemma 3.8.4. Let the integers \( a, c \) have images \( \bar{a},\bar{c} \) in \( \mathbb{Z}/N\mathbb{Z} \) . Then the following are equivalent:\n\n(1) \( \left\lbrack \frac{\bar{a}}{c}\right\rbrack \) has a lift \( \left\lbrack \frac{{a}^{\prime }}{{c}^{\prime }}\right\rbrack \in {\mathbb{Z}}^{2} \) with \( \gcd \left(... | Proof. If condition (1) holds then \( k\left( {a + {sN}}\right) + l\left( {c + {tN}}\right) = 1 \) for some integers \( k, l, s, t \), and so \( {ka} + {lc} + \left( {{ks} + {lt}}\right) N = 1 \), giving condition (2).\n\nIf condition (2) holds then \( {ad} - {bc} + {kN} = 1 \) for some \( b, d \), and \( k \), and the... | Yes |
Proposition 3.8.5. Let \( \Gamma \) be a congruence subgroup of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) and let \( P \) be the parabolic subgroup of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right), P = \left\{ {\pm \left\lbrack \begin{array}{ll} 1 & j \\ 0 & 1 \end{array}\right\rbrack : j \in \mathbb{Z}}\right... | Proof. The map is well defined since if \( {\Gamma \alpha P} = \Gamma {\alpha }^{\prime }P \) then \( {\alpha }^{\prime } = {\gamma \alpha \delta } \) for some \( \gamma \in \Gamma \) and \( \delta \in P \), so that \( \Gamma {\alpha }^{\prime }\left( \infty \right) = {\Gamma \gamma \alpha \delta }\left( \infty \right)... | Yes |
Proposition 4.2.1. For any \( \gamma \in {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \), \n\n\[ \n\left( {{E}_{k}^{\bar{v}}{\left\lbrack \gamma \right\rbrack }_{k}}\right) \left( \tau \right) = {E}_{k}^{\overline{v\gamma }}\left( \tau \right) \n\] | Proof. Compute for any \( \gamma \in {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \), using the fact that \( \Gamma \left( N\right) \) is normal in \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) at the second and fourth steps, that \n\n\[ \n\left( {{E}_{k}^{\bar{v}}{\left\lbrack \gamma \right\rbrack }_{k}}\right) \le... | Yes |
Corollary 4.2.2. \( {E}_{k}^{\bar{v}} \in {\mathcal{M}}_{k}\left( {\Gamma \left( N\right) }\right) \) . | Proof. As a subseries of the series \( {E}_{k} \) analyzed in Exercise 1.1.4, \( {E}_{k}^{\bar{v}} \) is holomorphic on \( \mathcal{H} \), meeting condition (1) of Definition 1.2.3. Since every \( \gamma \in \Gamma \left( N\right) \) reduces to the identity matrix modulo \( N \), Proposition 4.2.1 here shows that \( \o... | No |
Theorem 4.2.3. The Fourier expansion of \( {G}_{k}^{\bar{v}}\left( \tau \right) \) for \( k \geq 3 \) and \( \bar{v} \in {\left( \mathbb{Z}/N\mathbb{Z}\right) }^{2} \) a point of order \( N \) is\n\n\[ \n{G}_{k}^{\bar{v}}\left( \tau \right) = \delta \left( {\bar{c}}_{v}\right) {\zeta }^{{\bar{d}}_{v}}\left( k\right) + ... | When \( N = 1 \), the Fourier expansion agrees with \( {G}_{k}\left( \tau \right) \) from Chapter 1 for \( k \) even and it cancels to zero when \( k \) is odd (Exercise 4.2.6). Clearly the \( n \) th Fourier coefficient is bounded by \( C{n}^{k} \), completing the proof that \( {E}_{k}^{\bar{v}} \) is a modular form. | No |
Proposition 4.3.1. Let \( {\widehat{G}}_{N} \) be the dual group of \( {G}_{N} \) . Then \( {\widehat{G}}_{N} \) is isomorphic to \( {G}_{N} \) . In particular, the number of Dirichlet characters modulo \( N \) is \( \phi \left( N\right) \) . | The two groups are noncanonically isomorphic, meaning that constructing an actual isomorphism from \( {G}_{N} \) to \( {\widehat{G}}_{N} \) involves arbitrary choices of which elements map to which characters. The groups \( {G}_{N} \) and \( {\widehat{G}}_{N} \) satisfy the orthogonality relations (Exercise 4.3.1),\n\n... | No |
Lemma 4.3.2. Let \( N \) be a positive integer. If \( N = 1 \) or \( N = 2 \) then every Dirichlet character \( \chi \) modulo \( N \) satisfies \( \chi \left( {-1}\right) = 1 \) . If \( N > 2 \) then the number of Dirichlet characters modulo \( N \) is even, half of them satisfying \( \chi \left( {-1}\right) = 1 \) an... | Proof. The result for \( N = 1 \) and \( N = 2 \) is clear. If \( N > 2 \) then \( 4\left| N\right| \) or \( p \mid N \) for some odd prime \( p \) . The nontrivial character modulo 4 takes -1 (mod 4) to -1, and for every odd prime \( p \) the character modulo \( p \) taking a generator \( g \) of \( {G}_{p} \) to a pr... | Yes |
Theorem 4.5.1. The Eisenstein series \( {G}_{k}^{\psi ,\varphi } \) takes the form\n\n\[ \n{G}_{k}^{\psi ,\varphi }\left( \tau \right) = \frac{{C}_{k}g\left( \bar{\varphi }\right) }{{v}^{k}}{E}_{k}^{\psi ,\varphi }\left( \tau \right) ,\n\]\n\nwhere \( {E}_{k}^{\psi ,\varphi } \) has Fourier expansion\n\n\[ \n{E}_{k}^{\... | For any positive integer \( N \) and any integer \( k \geq 3 \), let \( {A}_{N, k} \) be the set of triples \( \left( {\psi ,\varphi, t}\right) \) such that \( \psi \) and \( \varphi \) are primitive Dirichlet characters modulo \( u \) and \( v \) with \( \left( {\psi \varphi }\right) \left( {-1}\right) = {\left( -1\ri... | No |
Proposition 4.10.1. For any function \( a : G \rightarrow \mathbb{C} \), the associated sum of theta functions\n\n\[ \n{\Theta }_{k}^{a}\left( \gamma \right) = \mathop{\sum }\limits_{{\bar{v} \in G}}\left( {a\left( \bar{v}\right) + {\left( -1\right) }^{k}\widehat{a}\left( {-\bar{v}}\right) }\right) {\vartheta }_{k}^{\b... | The functional equation for the associated sum of Eisenstein series follows as before. The constant terms of the series \( {\vartheta }_{k}^{\bar{v}}\left( \gamma \right) \) are 0, so that \( {\Theta }_{k}^{a}\left( {\gamma r}\right) \) converges to 0 rapidly as \( r \rightarrow \infty \) . Consider the Mellin transfor... | Yes |
Theorem 4.10.2. Let \( N \) be a positive integer and let \( G = {\left( \mathbb{Z}/N\mathbb{Z}\right) }^{2} \) . For any function \( a : G \rightarrow \mathbb{C} \), let \( {G}_{k}^{a}\left( {\tau, s}\right) \) be the associated sum (4.46) of Eisenstein series. Then for any integer \( k \) and any point \( \tau = x + ... | \[ {\left( \pi /N\right) }^{-s}\Gamma \left( {\left| k\right| /2 + s}\right) {G}_{k}^{a}\left( {\tau, s - k/2}\right) ,\;\operatorname{Re}\left( s\right) > 1 \] has a continuation to the full \( s \) -plane that is invariant under \( s \mapsto 1 - s \) . The continuation is analytic for \( k \neq 0 \) and has simple po... | Yes |
Theorem 4.11.3. Let \( N \) be a positive integer and let \( \chi : {\left( A/NA\right) }^{ * } \rightarrow {\mathbb{C}}^{ * } \) be a character, extended multiplicatively to A. Define\n\n\[ \n{\theta }_{\chi }\left( \tau \right) = \frac{1}{6}\mathop{\sum }\limits_{{\bar{u} \in A/{NA}}}\chi \left( u\right) {\theta }^{\... | The desired transformation of \( {\theta }_{\chi } \) under \( {\Gamma }_{0}\left( {3{N}^{2}}\right) \) follows from (4.49) since \( {\theta }_{\chi } = \mathop{\sum }\limits_{\bar{u}}\chi \left( u\right) {\theta }^{\bar{u}}{\left\lbrack \delta \right\rbrack }_{1} \) (Exercise 4.11.4). To finish proving the theorem not... | No |
Lemma 5.1.1. Let \( \Gamma \) be a congruence subgroup of \( {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) and let \( \alpha \) be an element of \( {\mathrm{{GL}}}_{2}^{ + }\left( \mathbb{Q}\right) \) . Then \( {\alpha }^{-1}{\Gamma \alpha } \cap {\mathrm{{SL}}}_{2}\left( \mathbb{Z}\right) \) is again a congruence sub... | Proof. There exists \( \widetilde{N} \in {\mathbb{Z}}^{ + } \) satisfying the conditions \( \Gamma \left( \widetilde{N}\right) \subset \Gamma ,\widetilde{N}\alpha \in \) \( {\mathrm{M}}_{2}\left( \mathbb{Z}\right) ,\widetilde{N}{\alpha }^{-1} \in {\mathrm{M}}_{2}\left( \mathbb{Z}\right) \) . Set \( N = {\widetilde{N}}^... | Yes |
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