arXiv subjects
Publications and source records attributed to Simon Kurmann.
We explain how to determine equations describing the ramification of an outer simple linear projection of a projective scheme in a way suited for explicit computations.
Consider the projective variety $X_λ$ of binary forms of degree $d$ whose linear factors are distributed according to the partition $λ$ of $d$. We determine minimal sets of local generators of the fiber product of $X_λ$ with its normalization, and we show that the local Jacobian matrices of this product contain the product of the identity matrix of maximal rank with a unit. We use this to fill a gap in a crucial proof in Chipalkatti's "On equations defining Coincident Root Loci". Also, we give a new description of the singular locus of $X_λ$ and a criterion for the smoothness of $X_λ$.
For any $k \in \Nat$, we show that the cone of $(k+1)$-secant lines of a closed subscheme $Z \subset \mathbb{P}^n_K$ over an algebraically closed field $K$ running through a closed point $p \in \mathbb{P}^n_K$ is defined by the $k$-th partial elimination ideal of $Z$ with respect to $p$. We use this fact to give an algorithm for computing secant cones. Also, we show that under certain conditions partial elimination ideals describe the length of the fibres of a multiple projection in a way similar to the way they do for simple projections. Finally, we study some examples illustrating these results, computed by means of {\sc Singular}.