Small rational curves on the moduli space of stable bundles (Q2909482)

Let \(C\) be a smooth, projective curve over \(\mathbb{C}\) of genus \(g\geq 2\) and let \([\mathcal{L}]\in \mathrm{Pic}^1(C)\). Let \(M=\mathcal{SU}_C(r,\mathcal{L})\) be the moduli space of stable vector bundles of rank \(r\) and determinant isomorphic to \(\mathcal{L}\). It is a smooth, projective variety of dimension \((r^2-1)(g-1)\). The present paper is devoted to the study of small rational curves on \(M\).NEWLINENEWLINEAs shown by Drezet, Narasimhan and Ramanan, \( \mathrm{Pic} M=\mathbb{Z}\cdot\Theta\), where \(\Theta\) is an ample generator, and \(-K_M=2\Theta\). Hence the Fano index of \(M\), i.e., the largest integer \(d\geq 1\), such that \(-K_M=dD\), \(D\in \mathrm{Pic} M\), is two.NEWLINENEWLINEIf \(X\) is a Fano variety of index \(d\) and \(l\subset X\) is a rational curve, we say that \(l\) \textit{has degree} \(k\) if \(-K_X\cdot l=kd\). We say that \(l\) is \textit{a line}, if it has degree one. In particular, for a rational curve \(\phi:\mathbb{P}^1\to M\), the degree is simply \(\deg (\phi^\ast\Theta)\).NEWLINENEWLINE\textit{M. S. Narasimhan} and \textit{S. Ramanan} [in: Tata Inst. fundam. Res., Stud. Math. 8, 291--345 (1978; Zbl 0427.14002)] showed that \(M\) is covered by rational curves of degree \(r\), the so-called \textit{Hecke curves}. \textit{S. Ramanan} [Math. Ann. 200, 69--84 (1973; Zbl 0239.14013)] found a family of lines contained in a proper closed subset of \(M\), and these were shown to be the only lines on \(M\) [\textit{X. Sun}, Math. Ann. 331, No. 4, 925--937 (2005; Zbl 1115.14027)].NEWLINENEWLINERational curves on \(M\) of degree smaller than \(r\) are known as \textit{small rational curves}. A result of \textit{X. Sun} (Theorem in [Zbl 1115.14027]) implies that curves of minimal degree passing through a general point of \(M\) are Hecke, and hence all small rational curves lie in a proper closed subset. This subset is known to have \((r-1)\) irreducible components [\textit{I. Choe}, Bull. Korean Math. Soc. 48, No. 2, 377--386 (2011; Zbl 1237.14040)].NEWLINENEWLINEThe present paper gives an estimate for the codimension of the locus of small rational curves (Section 2) and an explicit construction of all small rational curves in the case \(r=3\) (Section 3).NEWLINENEWLINEIn [Zbl 1115.14027], \textit{X. Sun} derived the following a formula for the degree of a rational curve \(\phi:\mathbb{P}^1\to M\). Suppose \(\phi\) is defined by a vector bundle \(E\) over \(X=C\times\mathbb{P}^1\). Let \(f= \mathrm{pr}_1\) and \(\pi= \mathrm{pr}_2\) be the two canonical projections and let the \textit{generic splitting type} of \(E\) be \(\alpha=(\alpha_1^{\oplus r_1},\dots, \alpha_n^{\oplus r_n})\in \bigoplus_i\mathbb{N}^{r_i}\). We may assume, after twisting, that \(\alpha_1>\dots >\alpha_n=0\). The bundle \(E\) admits a relative Harder-Narasimhan filtration \(E^{HN}_\bullet\), and the associated graded \(Gr^{HN}_\bullet(E)\) has torsion-free summands \(Gr^{HN}_i(E_\bullet):=E_i/E_{i-1}\) of generic splitting type \((\alpha_i^{\oplus r_i})\). Then Sun's formula states that NEWLINE\[NEWLINE \deg (\phi^\ast\Theta)= r\left(\sum_{i=1}^n c_2(F_i')+\sum_{i=1}^{n-1}(\mu(E)-\mu(E_i))(\alpha_i-\alpha_{i+1}) \mathrm{rk} E_i\right). NEWLINE\]NEWLINE Here \(F_i'=Gr^{HN}_i(E_\bullet)\otimes\pi^\ast\mathcal{O}(-\alpha_i)\), and \(\mu(E_i)\) is the slope of the restriction to a generic fibre of \(\pi\). Sun shows that for a small rational curve \(c_2(F_i')=0\) and \(F_i'=f^\ast V_i\), for some locally free sheaf \(V_i\) on \(C\).NEWLINENEWLINEIn Section 2, Lemma 2.1 the author uses Sun's formula to show that \(V_i\) is semi-stable of degree zero for \(i\leq n-1\), whlie \(V_n\) is stable of degree one. The bundle \(E\) is then obtained by taking successive extensions of \(f^\ast V_i\otimes \pi^\ast \mathcal{O}_{\mathbb{P}^1}(\alpha_i)\).NEWLINENEWLINENext the author goes on to estimate the codimension of the locus of small rational curves. In Theorem 2.4 it is proved that any small rational curve in \(M\) lies in a closed subset NEWLINE\[NEWLINE S = \bigcup_{0<r_1<\dots <r_n=r}S_{r_1\dots r_n} NEWLINE\]NEWLINE of codimension at least NEWLINE\[NEWLINE \min_{0<r_1<\dots <r_n=r}\left\{ \sum_{i=2}^n r_{i-1}(r_i-r_{i-1})(g-2)+ r_{n-1}(r_n-r_{n-1}-1) \right\} NEWLINE\]NEWLINE where \(0<r_1<\dots <r_n=r\) runs over \(n\) positive integers \(r_i\), satisfying \(\sum^{n-1}_{i=1}r_i(\alpha_i-\alpha_{i-1})<r\) for some \(n\) and some integers \(\alpha_1>\dots >\alpha_n\).NEWLINENEWLINESection 3 is devoted to the description of all small rational curves in \(M=\mathcal{SU}_C(3,\mathcal{L})\) of degree two: a small rational curve must have degree \(1\) or \(2\), and the case of lines is treated in [Zbl 1115.14027], [\textit{N. Mok} and \textit{X. T. Sun}, Sci. China, Ser. A 52, No. 4, 617--630 (2009; Zbl 1194.14052)]. Hence Sun's formula reduces to NEWLINE\[NEWLINE 2 =\sum_{i=1}^{n-1} \mathrm{rk}E_i (\alpha_i-\alpha_{i+1}), NEWLINE\]NEWLINE and there are only two possibilities for the bundle \(E\to C\times\mathbb{P}^1\):NEWLINENEWLINE(A) it fits in an extension of \(f^\ast V_2 \) by \( f^\ast V_1\otimes\pi^\ast\mathcal{O}_{\mathbb{P}^1}(2) \) with \(\mathrm{rk}V_1=1\), \(\mathrm{rk}V_2=2\), \(\deg V_1=0\), \(\deg V_2=1\)NEWLINENEWLINE (B) it fits in an extension of \(f^\ast V_2 \) by \(f^\ast V_1\otimes\pi^\ast\mathcal{O}_{\mathbb{P}^1}(1)\) with \(\mathrm{rk}V_1=2\), \( \mathrm{rk}V_2=1\), \(\deg V_1=0\), \(\deg V_2=1\).NEWLINENEWLINETo classify these one uses the description of universal extensions from [Zbl 0239.14013], Lemmas 2.3 and 2.4. See also [\textit{M. S. Narasimhan} and \textit{S. Ramanan}, Ann. Math. (2) 89, 14--51 (1969; Zbl 0186.54902), Proposition 3.1] and [\textit{M. S. Narasimhan} and \textit{C. S. Seshadri}, Ann. Math. (2) 82, 540--567 (1965; Zbl 0171.04803), Lemma 3.1]:NEWLINENEWLINEProposition [Ramanan, Zbl 0239.14013]. Let \((W_t)_{t\in T}\), \((V)_{t\in T}\) be two families of vector bundles parametrised by a variety \(T\), such that\newline \(\dim H^1(C,\underline{Hom}(V_t,W_t))\) is independent of \(t\in T\). Let \(G=R^1p_{T\ast}\underline{Hom}(V,W)\), and let \(\pi:\mathbb{P}(G)\to T\) be the corresponding projective bundle. Assume that \(H^i(T, p_{T\ast}\underline{Hom}(V,W)\otimes G^\vee)=0\), \(i=1,2\). Then the extension NEWLINE\[NEWLINE 0\longrightarrow \pi^\ast p_{C\times T}^\ast W\otimes p_{\mathbb{P}(G)}^\ast\mathcal{O}_{\mathbb{P}(G)}(1)\longrightarrow \mathcal{E}\longrightarrow \pi^\ast p_{C\times T}^\ast V\longrightarrow 0 NEWLINE\]NEWLINE is a family of bundles on \(C\), parametrised by \(\mathbb{P}(G)\). For each \(t\in T\) the restriction of \(\mathcal{E}\) to \(C\times \mathbb{P}(G_t)\) has the property that for any \(x\in \mathbb{P}(G_t)\) \(\left. E\right|_{C\times \{x\}}\) is an extension of \(V_t\) by \(W_t\) with extension class in the line \(x\).NEWLINENEWLINEFor dealing with vector bundles of type (A) or ones of type (B) with \(V_1\) unstable, one considers an extension \(E\) of \(V\) by \(\xi\) with extension class in the line \([e]\subset H^1(C,V^\vee\otimes C)\), where \([\xi]\in \mathrm{Pic}^1 C\) and \(\mathrm{rk}V=2\). The appropriate universal extension parametrising the data of all such \(x=([\xi],[V],[e])\) is constructed as follows.NEWLINENEWLINELet \(U_C(2,1)\) be the moduli space of stable vector bundles of rank two and degree one on \(C\), and let \(\mathcal{V}\to C\times U_C(2,1)\) be the universal family. Let \(J_C=\mathrm{Pic}^0 C\), \(J^1_C=\mathrm{Pic}^1 C\) and let \(\mathfrak{L}\to C\times J_C\) be the Poincaré line bundle. Denote by \(\mathcal{R}\subset J_C\times U_C(2,1)\) the subvariety of pairs \(([\xi],[V])\) with \(\det V\otimes \xi\simeq \mathcal{L}\). We set \(G:=R^1p_\ast (\mathcal{V}^\vee\otimes\mathfrak{L})\), where \(p=\mathrm{pr}_2:C\times \mathcal{R}\to\mathcal{R}\), and \(q: \mathcal{P}=\mathbb{P}(G)\to \mathcal{R}\). Then the universal extension is Ramanan's bundle \(\mathcal{E}\to C\times \mathcal{P}\) from above (setting \(T=\mathcal{P}\) and \(\pi=q\)).NEWLINENEWLINEThe bundle \(\mathcal{E}\) gives rise to a morphism \(\Phi: \mathcal{P}\to M=\mathcal{SU}_C(3,\mathcal{L})\).NEWLINENEWLINESimilarly, let \(U_C(2,0)\) be the coarse moduli space of semi-stable rank 2 bundles of degree zero, and \(U_C^{s}(2,0)\) the open set of stable bundles. Let \(\mathcal{R}'\subset U_C^{s}(2,0)\times J_C^1\) be the closed subvariety of pairs \(([V_1],[\xi'])\) with \(\det V_1\otimes \xi'=\mathcal{L}\). If \(\mathcal{V}\to C\times U_C(2,1)\) denotes the universal family, one constructs, using Hecke transformations, a family \(K(\mathcal{V})\to C\times \mathbb{P}(\mathcal{V}^\vee)\), giving a surjective morphism \(\theta: \mathbb{P}(\mathcal{V}^\vee)\to U_C(2,0)\). We also set \(\mathbb{P}(\mathcal{V}^\vee)^{s}:=\theta^{-1}(U_C^{s}(2,0))\) and \(T:= \left(\mathbb{P}(\mathcal{V}^\vee)^{s}\times J^1_C\right)\times_{U_C^{s}(2,0)\times J^1_C}\mathcal{R}'\). If \(p=\mathrm{pr}_2:C\times T\to T\) and \(\mathcal{F}=R^1p_\ast (\mathfrak{L}^\vee\otimes K(\mathcal{V}))\), let \(q':\mathcal{P}'=\mathbb{P}(\mathcal{F})\to T\). Then there exists a vector bundle \(\mathcal{E}\to C\times \mathcal{P}'\) which is an extension of \((1_C\times q')^\ast \mathfrak{L} \) by \((1_C\times q')^\ast K(\mathcal{V})\otimes \mathcal{O}_{\mathcal{P}'}(1) \) with the following property. Let \(x=\left([0\to V_1^\vee\to \mathcal{V}\to \mathcal{O}_p\to 0], [\xi], [\iota]\right)\in\mathcal{P}'\) so \(\det V_1\otimes \xi\simeq \mathcal{L}\), \([\iota]\in \mathbb{P}H^1(C,\xi^\vee\otimes V_1)\) and \([0\to V_1^\vee\to \mathcal{V}\to \mathcal{O}_p\to 0]\in \mathbb{P}(\mathcal{V}^\vee)^s\). Then the restriction \(\left. \mathcal{E}'\right|_{C\times \{x\}}\) is isomorphic to an extension \(E'\) of \(\xi\) by \(V_1\) with extension class in \([\iota]\).NEWLINENEWLINEThe author shows (Lemma 3.5) that \(\mathcal{E}'\) gives rise to a family of stable rank 3 bundles with determinant \(\mathcal{L}\), and hence to a morphism \(\Psi: \mathcal{P'}=\mathbb{P}(\mathcal{F})\to M=\mathcal{SU}_C(3,\mathcal{L})\). Finally, in Theorem 3.7 the author proves that there exist small rational curves on \(M\), and that each of them can be obtained in one of four possible ways: as the image under \(\Phi\) of a 1) rational curve of degree 2 in the fibre of \(q\) 2) double cover of a line in the fibre of \(q\) 3) line which is not in the fibre of \(q\) and maps to a line in \(U_C(2,\mathcal{L}')\), \([\mathcal{L}']\in \mathrm{Pic}^1 C\) or 4) as the image under \(\Psi\) of a line in \(\mathcal{P}'\) in the fibre of \(q'\).