Abstract
With any surjective rational map \(f: \mathbb {P}^n \dashrightarrow \mathbb {P}^n\) of the projective space, we associate a numerical invariant (ML degree) and compute it in terms of a naturally defined vector bundle \(E_f \longrightarrow \mathbb {P}^n\).
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With a collection of effective divisors \(D_0,\ldots ,D_m\) in the projective space \(\mathbb {P}:= \mathbb {C}\mathbb {P}^n\) is associated the maximum likelihood degree \((-1)^ne_{\text {top}}(\mathbb {P}{\setminus } D)\), \(D := \bigcup _{i=0}^m D_i\). Alternatively, letting \(\Omega ^1_{\mathbb {P}}(\log D)\) be the Saito’s sheaf, i.e., the double dual of the sheaf of logarithmic differential 1-forms, one computes the ML degree as the top Chern class \(c_n(\Omega ^1_{\mathbb {P}}(\log D))\) (we refer to [2, 4] for basic properties of ML degree, its connections with algebraic statistics, topology of arrangements, combinatorics, etc.). Note, however, that it is difficult to compute \(c_n(\Omega ^1_{\mathbb {P}}(\log D))\) in general (when D is not SNC).
In the present note, we study the ML degree under the condition that defining polynomials \(f_i\) of \(D_i\), \(0 \le i \le m = n\), span the linear system of a surjective rational map \(f: \mathbb {P}\dashrightarrow \mathbb {P}\) (see [1] and [6] for some aspects of such maps). Our main result (proved along the lines that follow) is the next.
In the previous setting, the ML degree \(c_n(\Omega ^1_{\mathbb {P}}(\log D))\) is equal to the coefficient of \(z^n\) in \(\frac{(1 - z\mathcal {O}_{\mathbb {P}}(1))^{n+1}}{\prod _{i=0}^n(1 - z\mathcal {O}_{\mathbb {P}}(D'_i))}\), where \(\bigcup _{i=0}^n D'_i =: D_{\text {red}}\) is the reduced scheme associated with D (so that \(D = D_{\text {red}}\) as sets).
For a vector bundle E over \(\mathbb {P}\), given by an affine open cover \(\mathbb {P}= \displaystyle \cup _{\alpha }\,U_{\alpha }\) and transition functions \(g_{\alpha \beta }: U_{\alpha } \cap U_{\beta } \longrightarrow \text {GL}(r,\mathbb {C})\), the pullback \(f^*(E)\) on \(\mathbb {P}{\setminus }{\{\Sigma :=\ \text {base locus of}\, f\}}\) is defined as usual (due to the surjectivity of f), via \(f^{-1}(U_{\alpha })\) and \(f^*(g_{\alpha \beta })\). Note that all \(f^{-1}(U_{\alpha })\) are affine open in \(\mathbb {P}\). Let \(\displaystyle \cup _{k}\,U_{\alpha ,k}\) be an affine open cover of \(f^{-1}(U_{\alpha })\), such that \(\mathbb {P}= \displaystyle \cup _{\alpha ,k}\,U_{\alpha ,k}\). Then, since \(\text {codim}\,\Sigma > 1\) and \(f^*(g_{\alpha \beta })\) are algebraic, every \(f^*(g_{\alpha \beta })\) extends through \(U_{\alpha ,k} \cap U_{\beta ,m} \cap \Sigma \) to each \(U_{\alpha ,k} \cap U_{\beta ,m}\). Furthermore, the 1-cocyle property of \(f^*(g_{\alpha \beta })\) (considered on \((\displaystyle \cup _{k}\,U_{\alpha ,k}) \cap (\displaystyle \cup _{m}\,U_{\beta ,m}) \supseteq f^{-1}(U_{\alpha }) \cap f^{-1}(U_{\beta })\)) is preserved and one gets a vector bundle, over \(\mathbb {P}\), which we denote again by \(f^*(E)\).
Furthermore, let \(x_0,\ldots ,x_n\) be projective coordinates on \(\mathbb {P}\), such that \(f^*(x_i) = f_i\). Denote by H the union of coordinate hyperplanes \(H_i := (x_i = 0) \subset \mathbb {P}\). There is an exact sequence
(see, e.g., [2, Lemma 2]). We have \(f^*(\mathcal {O}_{\mathbb {P}}) = \mathcal {O}_{\mathbb {P}}\) and \(f^*(\mathcal {O}_{\mathbb {P}}(H)) = \mathcal {O}_{\mathbb {P}}(D)\) by construction. Then, (2) pulls back to an exact sequence
Note, however, that the morphism \(\psi _D := f^*(\psi _H)\) (resp. \(\varphi _D := f^*(\varphi _H)\)) need not be injective (resp. surjective)—see below.
\(f^*(\Omega ^1_{\mathbb {P}}(\log H)) = \Omega ^1_{\mathbb {P}}(\log D)\).
FormalPara ProofThe bundle \(\Omega ^1_{\mathbb {P}}(\log H)\) (resp. \(\Omega ^1_{\mathbb {P}}(\log D)\)) is trivial over an affine open set not containing H (resp. D). Hence, as \(f^*(\mathcal {O}_{\mathbb {P}}) = \mathcal {O}_{\mathbb {P}}\), it suffices to restrict to an affine open \(U \subset \mathbb {P}^n\) (resp. \(f^{-1}(U)\)), such that \(U \cap H \ne \emptyset \) (we may also assume that \(x_0 \ne 0\) on U). Then, \(\Omega ^1_{\mathbb {P}}(\log H)\big \vert _U\) is generated by the local sections \(\sum _{i = 1}^n c_i\log x_i\), \(c_i \in \mathbb {C}\), whereas \(f^*(\Omega ^1_{\mathbb {P}}(\log H))\big \vert _{f^{-1}(U)}\) is generated by \(\sum _{i = 1}^n c_i\log f^*(x_i)\) (as usual we take double duals when needed). This yields \(f^*(\Omega ^1_{\mathbb {P}}(\log H))\big \vert _{f^{-1}(U)} = \Omega ^1_{\mathbb {P}}(\log D)\big \vert _{f^{-1}(U)}\) and the result follows. \(\square \)
Before finding \(f^*(\Omega ^1_{\mathbb {P}})\), we need an auxiliary construction. Namely, put \(d_f := \deg f_i\) and consider the subspace \(V \subset H^0(\mathbb {P},\mathcal {O}_{\mathbb {P}}(d_f))\) spanned by \(f_0,\ldots ,f_n\). Recall that by Kodaira’s construction of rational maps via linear systems, every point \(p \in f(\mathbb {P}{\setminus }{\Sigma })\) is represented by hyperplane \(H_p \subset V\), which consists of all polynomials from V vanishing at \(f^{-1}(p)\). Then, since f is surjective, this defines a vector bundle \(E_f \longrightarrow \mathbb {P}= f(\mathbb {P}{\setminus }{\Sigma })\), with fibers \(E_{f,p} = H_p\) for all p, and an exact sequence
for some line bundle \(\mathcal {L}\). It is easy to prove (by induction on n) that \(\mathcal {L} = \mathcal {O}_{\mathbb {P}}(-n-1)\). This implies that both \(E_f\) and \(f^*(E_f)\) are generated by global sections.
We now prove the following (“Hurwitz-type”):
\(f^*(\Omega ^1_{\mathbb {P}}) \subseteq \Omega ^1_{\mathbb {P}}\otimes _{\mathcal {O}_{\mathbb {P}}}\mathcal {O}_{\mathbb {P}}(-d_f + 1)\).
FormalPara ProofEach global section of \(f^*(E_f)\) is given by some choice of a basis (\(=\{x_0,\ldots ,x_n\}\)) in \(\mathbb {C}^{n+1}\) and a way every \(p \in \mathbb {P}\) (identified with \(\sum p_i x_i\) for \(p_i \in \mathbb {C}\)) is represented by a point in \(V \simeq \mathbb {C}^{n+1} = H^0(\mathbb {P},E_f)\). This yields a surjection
of vector bundles generated by global sections.
Now, observe that \(E_f \simeq T_{\mathbb {P}}\ (= \text {the dual of}\ \Omega ^1_{\mathbb {P}})\) by (5) and [5, Theorem 3.1]. Hence, \(f^*(\Omega ^1_{\mathbb {P}})\) embeds into \(\Omega ^1_{\mathbb {P}}\otimes _{\mathcal {O}_{\mathbb {P}}}\mathcal {O}_{\mathbb {P}}(-d_f + 1)\) by duality. \(\square \)
Note that \(\Omega ^1_{\mathbb {P}}(\log D) = \Omega ^1_{\mathbb {P}}(\log D_{\text {red}})\) (cf. the proof of Lemma 4). Hence, \(\varphi _D(\Omega ^1_{\mathbb {P}}(\log D)) = \bigoplus _{i = 0}^n \mathcal {O}_{D'_i}\). Further, it follows from (3) and Lemma 6 that the kernel of \(\varphi _D\) is a subsheaf of \(\psi _D(\Omega ^1_{\mathbb {P}} \otimes \mathcal {O}_{\mathbb {P}}(-d_f + 1))\), whose general local section is easily seen (by restricting on \(\mathbb {P}{\setminus }\Sigma \)) to coincide with a holomorphic 1-form, which vanishes at most on \(D_{\text {red}}\). One actually finds that this is a subbundle of \(\Omega ^1_{\mathbb {P}}\) generated by all such 1-forms. Thus, we get \(\text {Ker}\,\varphi _D = \Omega ^1_{\mathbb {P}}\) and an exact sequence
Taking the total Chern class of the latter concludes the proof of Theorem 1.
We summarize that any f defines, canonically, a fiberwise non-degenerate element \(e \in \text {Hom}\,(\mathbb {P}; T_{\mathbb {P}}, T_{\mathbb {P}} \otimes \mathcal {O}_{\mathbb {P}}(d_f - 1))\). This can also be seen as follows. Namely, the embedding \(\mathcal {L} \subset \mathbb {C}^{n+1}\) in (5) is given by some global sections \(s_0,\ldots ,s_n \in H^0(\mathbb {P},\mathcal {O}_{\mathbb {P}}(n + 1))\), so that \(x_i \mapsto s_i\), \(0 \le i \le n\), defines a regular surjective self-map of \(\mathbb {P}\). This yields a family (a “field”) \(\{H_p\}\) of hyperlines on \(\mathbb {P}\ni p\). After choosing e, one gets another family \(\{H'_p\}\), where \(H'_p \simeq H_p\) are spaces of forms of degree \(d_f\) and \(\displaystyle \bigcup H'_p = V\). Identify \(H'_p\) with the set of corresponding hypersurfaces that vanish at p. The map f is now obtained by sending each \(p \in H'_p\) to \(H_p\) (it is defined exactly on \(\mathbb {P}{\setminus }\displaystyle \bigcap H'_p\)). One thus obtains a description of the moduli spaces of surjective maps f. It would be interesting to relate this picture with [3], where the moduli of degree k rational self-maps of \(\mathbb {P}^1\) were interpreted as the moduli of (pairs of) monopoles, having magnetic charge k.
FormalPara Example 8The need for \(D_{\text {red}}\) in Theorem 1 is justified by the Frobenius map f, given by \(f_i := x_i^{d_f}\), \(0 \le i \le n\); ML degree of f equals \((-1)^ne_{\text {top}}((\mathbb {C}^*)^n) = \mathbf{0}\) in this case. Furthermore, one computes the ML degree of f in [6, Example1.6] to be \(\mathbf{9}\), which can also be seen directly from [2, Corollary 6] (here, the divisors \(D_i\) satisfy the GNC condition). Indeed, in this case, \(D_i\) are reduced and \(\deg f_i = 2\) for all i, so that the expression with Chern classes from Theorem 1 becomes
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I am grateful to anonymous referee for helpful comments and corrections.
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Karzhemanov, I. Maximum Likelihood Degree of Surjective Rational Maps. Arnold Math J. 8, 513–516 (2022). https://doi.org/10.1007/s40598-022-00207-0
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DOI: https://doi.org/10.1007/s40598-022-00207-0