Abstract
This note continues our previous work on special secant defective (specifically, conic connected and local quadratic entry locus) and dual defective manifolds. These are now well understood, except for the prime Fano ones. Here we add a few remarks on this case, completing the results in our papers (Russo in Math Ann 344:597–617, 2009; Ionescu and Russo in Compos Math 144:949–962, 2008; Ionescu and Russo in J Reine Angew Math 644:145–157, 2010; Ionescu and Russo in Am J Math 135:349–360, 2013; Ionescu and Russo in Math Res Lett 21:1137–1154, 2014); see also the recent book (Russo, On the Geometry of Some Special Projective Varieties, Lecture Notes of the Unione Matematica Italiana, Springer, 2016).
Similar content being viewed by others
References
Ballico, E., Chiantini, L.: On smooth subcanonical varieties of codimension 2 in \({\mathbb{P}}^n\), \(n\ge 4\). Ann. Math. Pura Appl. 135, 99–117 (1983)
Barth, W., Larsen, M.E.: On the homotopy groups of complex projective algebraic manifolds. Math. Scand. 30, 88–94 (1972)
Beltrametti, M.C., Fania, M.L., Sommese, A.J.: On the discriminant variety of a projective manifold. Forum Math. 4, 529–547 (1992)
Bonavero, L., Höring, A.: Counting conics in complete intersections. Acta Math. Vietnam. 35, 23–30 (2010)
Ein, L.: Varieties with small dual variety. I. Invent. Math. 86, 63–74 (1986)
Ein, L.: Varieties with small dual variety. II. Duke Math. J. 52, 895–907 (1985)
Fu, B.: Inductive characterizations of hyperquadrics. Math. Ann. 340, 185–194 (2008)
Fujita, T.: Classification Theories of Polarized Varieties, London Mathematical Society Lecture Note Series, vol. 155. Cambridge University Press (1990)
Hwang, J.M.: Geometry of minimal rational curves on Fano manifolds. In: School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000), ICTP Lecture Notes, vol. 6. Abdus Salam Int. Cent. Theoret. Phys., pp. 335–393 (2001)
Hwang, J.M., Kebekus, S.: Geometry of chains of minimal rational curves. J. Reine Angew. Math. 584, 173–194 (2005)
Ionescu, P., Russo, F.: Varieties with quadratic entry locus. II. Compos. Math. 144, 949–962 (2008)
Ionescu, P., Russo, F.: Conic-connected manifolds. J. Reine Angew. Math. 644, 145–157 (2010)
Ionescu, P., Russo, F.: Manifolds covered by lines and the Hartshorne conjecture for quadratic manifolds. Am. J. Math. 135, 349–360 (2013)
Ionescu, P., Russo, F.: On dual defective manifolds. Math. Res. Lett. 21, 1137–1154 (2014)
Kaji, H.: Homogeneous projective varieties with degenerate secants. Trans. Am. Math. Soc. 351, 533–545 (1999)
Kobayashi, S., Ochiai, T.: Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ. 13, 31–47 (1973)
Kollár, J.: Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. vol. 32, (3) Springer (1996)
Mok, N.: Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents. In: Third International Congress of Chinese Mathematicians. Part 1, 2. AMS/IP Studies in Advanced Mathematics, vol. 42, pt. 1, 2, pp. 41–61 American Mathematical Society, Providence (2008)
Mori, S.: Projective manifolds with ample tangent bundle. Ann. Math. 110, 593–606 (1979)
Mukai, S.: Biregular classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Natl. Acad. Sci. USA 86, 3000–3002 (1989)
Muñoz, R.: Varieties with low dimensional dual variety. Manuscr. Math. 94, 427–435 (1997)
Novelli, C., Occhetta, G.: Projective manifolds containing a large linear subspace with nef normal bundle. Mich. Math. J. 60, 441–462 (2011)
Okonek, C., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces, Progress in Mathematics 3. Springer, US (1980)
Russo, F.: Varieties with quadratic entry locus. I. Math. Ann. 344, 597–617 (2009)
Russo, F.: On the Geometry of Some Special Projective Varieties, Lecture Notes of the Unione Matematica Italiana. Springer (2016)
Wiśniewski, J.: On a conjecture of Mukai. Manuscr. Math. 68, 135–141 (1990)
Zak, F.L.: Tangents and Secants of Algebraic Varieties, Transl. Math. Monogr., vol. 127. American Mathematical Society (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Alexandru Lascu.
Rights and permissions
About this article
Cite this article
Ionescu, P., Russo, F. Remarks on defective Fano manifolds. Ann Univ Ferrara 63, 133–146 (2017). https://doi.org/10.1007/s11565-017-0270-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11565-017-0270-6