Premise
Let $X$ be a projective variety of dimension $n\geq1$ over an algebraically closed field of characteristic $0$.
A Higgs sheaf $\mathfrak{E}$ is a pair $(E,\varphi)$ where $E$ is a $\mathcal{O}_X$-coherent sheaf equipped with a morphism $\varphi\colon E\to E\otimes\Omega^1_X$ such that the composition $$ \varphi\wedge\varphi\colon E\xrightarrow{\varphi}E\otimes\Omega^1_X\xrightarrow{\varphi\otimes\operatorname{Id}}E\otimes\Omega^1_X\otimes\Omega^1_X\to E\otimes\Omega^2_X $$ vanishes. A Higgs subsheaf of a Higgs sheaf $(E,\varphi)$ is a $\varphi$-invariant subsheaf $G$ of $E$, i.e., $\varphi(G)\subset~G\otimes~\Omega_X^1$. A Higgs quotient of $\mathfrak{E}$ is a quotient of $E$ such that the corresponding kernel is $\varphi$-invariant. A Higgs bundle is a Higgs sheaf whose underlying coherent sheaf is locally free of rank $r$.
Let $0<s<r$ an integer number. Let $p_s\colon\mathrm{Gr}_s(E)\to X$ be the Grassmann bundle parameterizing rank $s$ locally free quotients of $E$ (see [FW]). In the short exact sequence of vector bundles over $\mathrm{Gr}_s(E)$ $$ 0\to S_{r-s,E}\xrightarrow{\psi} p_s^{*}E\xrightarrow{\eta}Q_{s,E}\to0 $$ $S_{r-s,E}$ is the universal rank $r-s$ subbundle and $Q_{s,E}$ is the universal rank $s$ quotient bundle of $p_s^{*}E$. Let $\mathfrak{E}=(E,\varphi)$ be a rank $r$ Higgs bundle on $X$. One defines closed subschemes $\mathrm{HGr}_s(\mathfrak{E})\subset\mathrm{Gr}_s(E)$ (the Higgs-Grassmann scheme) as the zero loci of the composite morphisms $$ (\eta\otimes\text{Id})\circ p_s^{*}(\varphi)\circ\psi\colon S_{r-s,E}\to Q_{s,E}\otimes p_s^{*}\Omega_X^1. $$ The restriction of previous sequence to $\mathrm{HGr}_s(\mathfrak{E})$ yields a universal short exact sequence $$ 0\to\mathfrak{S}_{r-s,\mathfrak{E}}\xrightarrow{\psi}\rho_s^{*}\mathfrak{E}\xrightarrow{\eta}\mathfrak{Q}_{s,\mathfrak{E}}\to0, $$ where $\mathfrak{Q}_{s,\mathfrak{E}}=Q_{s,E}\vert_{\mathrm{HGr}_s(\mathfrak{E})}$ is equipped with the quotient Higgs field induced by $\rho_s^{*}\varphi$ (here $\rho_s=p_s\vert_{\mathrm{HGr}_s(\mathfrak{E})}\colon\mathrm{HGr}_s(\mathfrak{E})\to X$).
In Lemma A.7 of [B,GO] the authors wrote: "[...] Let $C$ be a curve in $\mathrm{HGr}_1\left(\mathfrak{Q}_{s,\mathfrak{E}}\right)$. Possibily after a base change, we may assume that $C$ projects isomorphically onto a curve $C^{\prime}$ in $\mathrm{HGr}_s(\mathfrak{E})$ and that this projects isomorphically onto $X$. [...]" under the hypothesis that $X$ is a smooth curve. Moreover, the numerical class of $C$ (as $1$-cycle) does not change.
Questions
- Why is all this (base change, projection onto another curve, invariance of numercal class) possible?
- Does it depend on dimension of $X$? I mean, does it work also when $X$ is smooth and $\dim X\geq2$?
Bibliography
[B,GO] U. Bruzzo, B. Graña Otero - Semistable and Numerically Effective Principal (Higgs) Bundles, Advances in Mathematics 226 (2011) 3655-3676.
[FW] W. Fulton (1998) Intersection theory. Second edition, Springer-Verlag.
