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Register a new userOn the Splitting Principle of Beniamino Segre
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We state and prove in modern terms a Splitting Principle first claimed by Beniamino Segre in 1938, which should be regarded as a strong form of the classical Principle of Connectedness.
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We study instanton bundles $E$ on $mathbb{P}^1times mathbb{P}^1 times mathbb{P}^1$. We construct two different monads which are the analog of the monads for instanton bundles on $mathbb P^3$ and on the flag threefold $F(0,1,2)$. We characterize the Gieseker semistable cases and we prove the existence of $mu$-stable instanton bundles generically trivial on the lines for any possible $c_2(E)$. We also study the locus of jumping lines.
We introduce the notion of r-th Terracini locus of a variety and we compute it for at most three points on a Segre variety.
This paper discusses a central theorem in birational geometry first proved by Eugenio Bertini in 1891. J.L. Coolidge described the main ideas behind Bertinis proof, but he attributed the theorem to Clebsch. He did so owing to a short note that Felix Klein appended to the republication of Bertinis article in 1894. The precise circumstances that led to Kleins intervention can be easily reconstructed from letters Klein exchanged with Max Noether, who was then completing work on the lengthy report he and Alexander Brill published on the history of algebraic functions [Brill/Noether 1894]. This correspondence sheds new light on Noethers deep concerns about the importance of this report in substantiating his own priority rights and larger intellectual legacy.
Special types of quartic surfaces were much studied objects during the 1860s. Quartics were thus very much in the air when Sophus Lie and Felix Klein first met in Berlin in 1869. As this study shows, such surfaces played a major role in their subsequent work, much of which centered on linear and quadratic line complexes. This mutual interest led them to a number of new results on the quartic surfaces of Steiner, Plucker, and Kummer, as well as various types of ruled quartics studied earlier by Cremona. This paper, which draws on unpublished archival sources as well as published work from the period 1869-1872, underscores the importance of this aspect of the early geometrical work of these two famous figures. A highlight was Lies line-to-sphere transformation, which led to surprising new findings on properties of asymptotic curves on Kummer surfaces.
A conjecture of Hirschowitzs predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle, i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartans equivalence method to a suitable differential system on the universal family of the Douady space of the complex manifold, we prove that this conjecture is true if $A$ is a Fano manifold, or if the global sections of $W$ separate points of $A$. Our method shows more generally that for any unobstructed compact submanifold $A$ in a complex manifold, if the normal bundle is globally generated and its sections separate points of $A$, then a sufficiently general deformation of $A$ satisfies the formal principle. In particular, a sufficiently general smooth free rational curve on a complex manifold satisfies the formal principle.
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