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Smooth complex polarized varieties $(X,L)$ with a vector subspace $V subseteq H^0(X,L)$ spanning $L$ are classified under the assumption that the locus ${Cal D}(X,V)$ of singular elements of $|V|$ has codimension equal to $dim(X)-i$, $i=3,4,5$, the last case under the additional assumption that $X$ has Picard number one. In fact it is proven that this codimension cannot be $dim(X)-4$ while it is $dim(X)-3$ if and only if $(X,L)$ is a scroll over a smooth curve. When the codimension is $dim(X)-5$ and the Picard number is one only the Plucker embedding of the Grassmannian of lines in $Bbb P^4$ or one of its hyperplane sections appear. One of the main ingredients is the computation of the top Chern class of the first jet bundle of scrolls and hyperquadric fibrations. Further consequences of these computations are also provided.
An MBM locus on a hyperkahler manifold is the union of all deformations of a minimal rational curve with negative self-intersection. MBM loci can be equivalently defined as centers of bimeromorphic contractions. It was shown that the MBM loci on defo
Let V be a smooth equidimensional quasi-affine variety of dimension r over the complex numbers $C$ and let $F$ be a $(ptimes s)$-matrix of coordinate functions of $C[V]$, where $sge p+r$. The pair $(V,F)$ determines a vector bundle $E$ of rank $s-p$
We present algebraic and geometric arguments that give a complete classification of the rational normal scrolls that are hyperplane section of a given rational normal scrolls.
Motivated by the realizability problem for principal tropical divisors with a fixed ramification profile, we explore the tropical geometry of the double ramification locus in $mathcal{M}_{g,n}$.There are two ways to define a tropical analogue of the
We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in P^3 whose function field has le