By Alexandru Buium
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This booklet offers the idea of quadratic and hermitian varieties over earrings in a really common surroundings. It avoids, so far as attainable, any limit at the attribute and takes complete benefit of the functorial homes of the idea. it isn't an encyclopedic survey. It stresses the algebraic points of the idea and avoids - is reasonably overlapping with different books on quadratic varieties (like these of Lam, Milnor-Husemöller and Scharlau).
This quantity is a homage to the reminiscence of the Spanish mathematician Federico Gaeta (1923-2007). except a old presentation of his lifestyles and interplay with the classical Italian college of algebraic geometry, the amount offers surveys and unique learn papers at the arithmetic he studied.
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1–16 The Lüroth Problem [B1] [B2] [B3] [B4] [B5] [B6] [B7] [BBK] [BCSS] [B-L] [Bl] [Bo] [B-S] [C] [C-G] [Ch] [Co] [C-O] [C-P1] [C-P2] [C-S] 25 A. Beauville, Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. Éc. Norm. Supér. 10, 309–391 (1977) A. Beauville, Les singularités du diviseur de la jacobienne intermédiaire de l’hypersurface cubique dans P4 , in Algebraic Threefolds (Cime, Varenna, 1981). Lecture Notes in Mathematics, vol. 947 (Springer, Berlin/New York, 1982), pp. 190–208 A. Beauville, Variétés rationnelles et unirationnelles, in Algebraic Geometry – Open Problems (Proc.
Another example was found by Tregub [Tre93]: Suppose there is a quartic Veronese surface V ' P2 X meeting P transversally at three points. Then its proper transform VQ XQ is a section of q, giving rationality. To generalize this, we employ a basic property of quadric surfaces due to Springer (cf. 1] and [Swa89]): Let Q P3K be a quadric surface smooth over a field K. L/ ¤ ;. K/ ¤ ; and Q is rational over K via projection from a rational point. This applies when there exists a surface W X intersecting the generic fiber of q transversally in an odd number of points.
Thus the Stein factorization Q 2 / ! S ! X=P yields a degree two K3 surface—the double cover S ! P2 branched over B—and Q 2 / ! S. 11]): • the generic fiber of q is rational over K; • q admits a rational section; • r admits a rational section. The resulting birational map 1 W X Ü P2 P2 blows down the lines incident to the section of q, which are parametrized by a surface birational to S. Cubic fourfolds containing a plane have been re-examined recently from the perspective of twisted K3 surfaces and their derived categories [Kuz10, MS12, Kuz17].