arXiv subjects
Publications and source records attributed to Mingshuo Zhou.
For a classical simple and simply connected group $G$, let $\mathcal{M}_{G,ω}$ be the moduli space of $ω$-semistable parabolic $G$-bundles on a complex smooth projective curve of genus $g$. We prove two results in this article: (1) $\mathcal{M}_{G,ω}$ is of Fano type when $g\geq 3$; (2) the algebra of conformal blocks on any $n$-pointed stable curve for a classical simple Lie algebra is finitely generated.
Let $C$ be a nonsingular projective curve over an algebraically closed field of characteristic $p>0$ and $I\subset C$ be a finite set. If $\mathcal{U}_{C,\,ω}$ denotes the moduli space of semistable parabolic bundles of rank $r$ and degree $d$ on $C$ with parabolic structures determined by $ω=(k,\{\vec n(x),\vec a(x)\}_{x\in I})$, we prove that $\mathcal{U}_{C,\,ω}$ is \textit{$F$-split} for generic $C$ and generic choice of $I$ when $p>3r$.
We prove two recurrence relations among dimensions $$D_g(r,d,ω):={\rm dim}\,{\rm H}^0(\mathcal{U}_{C,\,ω},Θ_{\mathcal{U}_{C,\,ω}})$$ of spaces of generalized theta functions on moduli spaces $\mathcal{U}_{C,\,ω}$. By using of these recurrence relations, an explicit formula (Verlinde formula) of $D_g(r,d,ω)$ is proved (See Theorem 4.3).
We prove moduli spaces of semistable parabolic bundles and generalized parabolic sheaves with fixed determinant on a smooth projective curve are globally $F$-regular type.
For a minimal smooth projective surface $S$ of general type over a field of characteristic $p>0$, we prove that $K^2_S\le 32χ(\cal{O}_S).$ Moreover, if $18χ(\cal{O}_S) 0$, which answers completely a question of Shepherd-Barron.
Let $X$ be a minimal surface of general type over an algebraically closed field $\mathbf{k}$ of $\mathrm{char}.(\mathbf{k})=p\ge 0$. If the Albanese morphism $a_X:X\to \mathrm{Alb}_X$ is generically finite onto its image, we formulate a constant $c(X,L)\ge 0$ for a very ample line bundle $L$ on $\mathrm{Alb}_X$ such that $c(X,L)=0$ if and only if $\dim \mathrm{Alb}_X=2$ and $a_X: X\to \mathrm{Alb}_X$ is a double cover. A refined Severi inequality $$K^2_X\ge (4+{\rm min}\{\,c(X,L),\,\frac{1}{3}\,\})χ(\mathcal{O}_X)$$ is proved. Then we prove that $K^2_X=4χ(\mathcal{O}_X)$ if and only if the canonical model of $X$ is a flat double cover of an Abelian surface.
We prove that moduli spaces of semistable parabolic bundles and generalized parabolic sheaves (GPS) with a fixed determinant on a smooth projective curve are globally F-regular type. As an application, we prove vanishing theorems on the moduli spaces of semistable parabolic sheaves on a singular curve, which combining with Factorization theorems in [24] and [25] give two recurrence relations among dimensions of spaces of generalized theta functions. By using of these recurrence relations, we prove an explicit formula (Verlinde formula) for the dimension of spaces of generalized theta functions.
We prove the slope inequality for a relative minimal surface fibration in positive characteristic via Xiao's approach. We also prove a better low bound for the slope of non-hyperelliptic fibrations.
Fix a smooth projetive curve $\mathcal {C}$ of genus $g\geq 2$ and a line bundle $\mathcal{L}$ on $\mathcal{C}$ of degree $d$. Let $M:= \mathcal{SU}_{\mathcal{C}}(r, \mathcal{L})$ be the moduli space of stable vector bundles on $\mathcal{C}$ of rank $r$ and with fixed determinant $\mathcal{L}$. We prove that any rational curve on $M$ is a generalized Hecke curve. Furthermore, we study the lines on $M$, and prove that $M$ is covered by the lines when $(r, d)=r$; for the case $(r,d) (r,d)+1$. Finally, we prove that there are no $(1,0)$-stable (resp., $(0,1)$-stable) bundles for $g=2$, $r=2$ and $d$ is odd as an application.
Let $X$ be a smooth projective surface over an algebraically closed field $k$ of characteristic $p> 0$ with $Ω_{X}^{1}$ semistable and $μ(Ω_{X}^{1})>0$. For any semistable (resp. stable) bundle $W$ of rank $r$, we prove that $F_*W$ is semistable (resp. stable) when $p\geq r(r-1)^2+1$.
Let X be a smooth projective curve over an algebraic closed field of characteristic p and F be the Frobenius morphism of X. Here, I give a negative answer to the guess that the length of the Harder-Narasimhan of F*W is not bigger than p, where W is a semistable bundle on X.
Let X be a smooth projective curve of genus $g\geq 2$ defined over an algebraically closed field k of characteristic $p>0$ and let $F:X\rightarrow X_{1}$ be the relative k-linear Frobenius map. We prove (Theorem 1.1) E is a stable bundle on $X_{1}$ with $I(E)= (p-1)(2g-2)$ if and only if E is the direct image of some stable bundle W on $X$.