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The prolongation g^{(k)} of a linear Lie algebra g subset gl(V) plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras g subset gl(V) with non-zero prolongations. If g is the Lie algebra aut(hat{S}) of infinitesimal linear automorphisms of a projective variety S subset BP V, its prolongation g^{(k)} is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this perspective, understanding the prolongation aut(hat{S})^{(k)} is useful in questions related to the automorphism groups of uniruled projective manifolds. Our main result is a complete classification of irreducible non-degenerate nonsingular variety with non zero prolongations, which can be viewed as a generalization of the result of Cartan and Kobayashi-Nagano. As an application, we show that when $S$ is linearly normal and Sec(S) eq P(V), the blow-up of P(V) along S has the target rigidity property, i.e., any deformation of a surjective morphism Y to Bl_S(PV) comes from the automorphisms of Bl_S(PV).
In his groundbreaking work on classification of singularities with regard to right and stable equivalence of germs, Arnold has listed normal forms for all isolated hypersurface singularities over the complex numbers with either modality less than or
We prove a structure theorem for projective varieties with nef anticanonical divisors.
We generalise Flo{}ystads theorem on the existence of monads on the projective space to a larger set of projective varieties. We consider a variety $X$, a line bundle $L$ on $X$, and a base-point-free linear system of sections of $L$ giving a morphis
Euler-symmetric projective varieties are nondegenerate projective varieties admitting many C*-actions of Euler type. They are quasi-homogeneous and uniquely determined by their fundamental forms at a general point. We show that Euler-symmetric projec
Let X be a nonsingular projective algebraic variety, and let S be a line bundle on X. Let A = (a_1,..., a_n) be a vector of integers. Consider a map f from a pointed curve (C,x_1,...,x_n) to X satisfying the following condition: the line bundle f*(S)