For $r>2$, we study the moduli space parameterising fixed degree morphisms $\mathbb P^r\to X$ where $X$ is a smooth hypersurface of low degree. More precisely, we prove the following result: let $n\geq 2, e\geq 1$ and $X\subset \mathbb P^{n-1}$ a smooth degree $d\geq 2$ hypersurface over an algebraically closed field of characteristic zero or greater than $d$, then $\mathrm{Mor}_e(\mathbb P^r, X)$ is irreducible of the expected dimension if \[ n>2^d(d-1)\binom{de+r-1}{r-1}. \] Our result extends the result of Browning-Yamagishi for $r=2$ via multiblock Weyl differencing.
Let $\F$ be a codimension-one holomorphic foliation of degree $d$ on $\PP^n$, $n\geq3$, admitting an invariant hyperplane $H$. We study the extremal situation in which $S=(H\cap\Sing(\F))_{\rm red}$ is an irreducible hypersurface of $H$ of degree $d+1$. When $d+1$ is a power of a prime, we prove that, in suitable homog...
Let $M^4\hookrightarrow\mathbb S^5(1)$ be a closed CMC hypersurface with constant scalar curvature and constant third power sum $f_3=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$. We prove that if $M^4$ has exactly two distinct principal curvatures at some point, then it is isoparametric. More precisely, it is a Clifford torus of th...
Let $X=\mathbb{P}^3$ and let \[ \alpha_n=(0,1,-\tfrac12,\tfrac16-n)\in H^{\mathrm{even}}(X,\mathbb{Q}) \] with respect to the basis $1,H,H^2,H^3$, where $H=c_1(\mathcal{O}_X(1))$. We prove that every Gieseker semistable sheaf on $X$ with Chern character $\alpha_n$ is stable and uniquely of the form $\iota_{P*}\mathcal{...
R. Anderson· Journal of Geometry and Phys...· 0 citations
In this paper, we study the hypersurfaces in $\mathbb{H}^m\times\mathbb{H}^n$ ($m\geq3, n\geq2$) with constant principal curvatures. Let $g$ be the number of distinct constant principal curvatures. First, we classify all such hypersurfaces with $g\leq2$. Then, we prove that a hypersurface with constant principal curvat...
Let $X:M^n\to\mathbb R^{n+1}$ be a complete, connected, two-sided minimal immersion without boundary, where $n=3,4$. We prove that finite $\delta$-index, finite Morse index, and finite total curvature are equivalent for $\delta>((n-1)/n)^2$, without assumptions on properness, volume growth, or topology. The main estima...
For $m=2,3$, we prove that every smooth immersion $F:\mathbb{R}\mathbb{P}^m\looparrowright\overline{\mathbb B}^{,N}(1)$ satisfies $\kappa(F)^2\ge 2m/(m+1)$, with equality only for the Veronese embedding, up to congruence. We also prove that a closed, connected, orientable three-manifold admitting an immersion into a Eu...
Tsz-Kiu Aaron Chow, Jingbo Wan· 0 citations
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