arXiv subjects
Publications and source records attributed to Lucas Mioranci.
We generalize techniques by Coskun, Riedl, and Yeong, and obtain an almost optimal bound on the degree for the algebraic hyperbolicity of very general hypersurfaces in rational homogeneous varieties. As examples, we work out the cases of very general hypersurfaces in Grassmannians and products therefore, orthogonal and symplectic Grassmannians, and flag varieties.
We determine all triples $(e,d,n)$ for which a general degree $d$ hypersurface $X\subset \mathbb{P}^n$ contains a degree $e$ rational curve $C$ with balanced restricted tangent bundle $T_X|_C$. In addition, we show how to compute explicit examples of hypersurfaces with balanced $T_X|_C$ when $C$ is a rational normal curve.
Let $C$ be the rational normal curve of degree $e$ in $\mathbb{P}^n$, and let $X\subset \mathbb{P}^n$ be a degree $d\ge 2$ hypersurface containing $C$. In previous work, I. Coskun and E. Riedl showed that the normal bundle $N_{C/X}$ is balanced for a general $X$. H. Larson studied the case of lines ($e=1$) and computed the dimension of the space of hypersurfaces for which $N_{C/X}$ has a given splitting type. In this paper, we work with any $e\ge 2$. We compute explicit examples of hypersurfaces for all possible splitting types, and for $d\ge 3$, we compute the dimension of the space of hypersurfaces for which $N_{C/X}$ has a given splitting type. For $d=2$, we give a lower bound on the maximum rank of quadrics with fixed splitting type.