arXiv subjects
Publications and source records attributed to Maria Gioia Cifani.
Let X be an irreducible, reduced complex projective hypersurface of degree d. A uniform point for X is a point P such that the projection of X from P has maximal monodromy. We extend and improve some results concerning the finiteness of the locus of non-uniform points for projections of hypersurfaces obtained by the authors and Cuzzucoli only for P not contained in X.
Let $S$ be a smooth algebraic surface in $\mathbb{P}^3(\mathbb{C})$. A curve $C$ in $S$ has a cohomology class $η_C \in H^1 \hspace{-3pt}\left( Ω^1_S \right)$. Define $α(C)$ to be the equivalence class of $η_C$ in the quotient of $H^1 \hspace{-3pt}\left( Ω^1_S \right)$ modulo the subspace generated by the class $η_H$ of a plane section of $S$. In the paper "Reconstructing subvarieties from their periods" the authors Movasati and Sertöz pose several interesting questions about the reconstruction of $C$ from the annihilator $I_{α(C)}$ of $α(C)$ in the polynomial ring $R=H^0_*(\mathcal{O}_{\mathbb{P}^3})$. It contains the homogeneous ideal of $C$, but is much larger as $R/I_{α(C)}$ is artinian. We give sharp numerical conditions that guarantee $C$ is reconstructed by forms of low degree in $I_{α(C)}$. We also show it is not always the case that the class $α(C)$ is \textit{perfect}, that is, that $I_{α(C)}$ could be bigger than the sum of the Jacobian ideal of $S$ and of the homogeneous ideals of curves $D$ in $S$ for which $I_{α(D)}=I_{α(C)}$.
Let $X$ be a general complex projective hypersurface in $\mathbb{P}^{n+1}$ of degree $d>1$. A point $P$ not in $X$ is called uniform if the monodromy group of the projection of $X$ from $P$ is isomorphic to the symmetric group. We prove that all the points in $\mathbb{P}^{n+1}$ are uniform for $X$, generalizing a result of Cukierman on general plane curves.
Let $X$ be an irreducible, reduced complex projective hypersurface of degree $d$. A point $P$ not contained in $X$ is called uniform if the monodromy group of the projection of $X$ from $P$ is isomorphic to the symmetric group $S_d$. We prove that the locus of non--uniform points is finite when $X$ is smooth or a general projection of a smooth variety. In general, it is contained in a finite union of linear spaces of codimension at least $2$, except possibly for a special class of hypersurfaces with singular locus linear in codimension $1$. Moreover, we generalise a result of Fukasawa and Takahashi on the finiteness of Galois points.