Abstract
Corrado Segre played a leading role in the foundation of line geometry. We survey some recent results on degeneracy loci of morphisms of vector bundles where he still is of profound inspiration.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
D. Bazan and E. Mezzetti, On the construction of some Buchsbaum varieties and the Hilbert scheme of elliptic scrolls in ℙ5, Geom. Dedicata 86 (2001), no. 1–3, 191–204.
A. Boralevi, D. Faenzi, and E. Mezzetti, Linear spaces of matrices of constant rank and instanton bundles, Adv. Math. 248 (2013), 895–920.
A. Boralevi, D. Faenzi, and P. Lella, Truncated modules and linear presentations of vector bundles, arXiv:1505.04204, 2015.
A. Boralevi and E. Mezzetti, Planes of matrices of constant rank and globally generated vector bundles, Ann. Inst. Fourier 65 (2015), no. 5, 2069–2089.
G. Castelnuovo, Ricerche di geometria della retta nello spazio a quattro dimensioni, Atti R. Istituto Veneto (1891), 855–901.
P. De Poi, Threefolds of ℙ5 with one apparent quadruple point, Comm. Algebra 31 (2003), no. 4, 1927–1947.
P. De Poi and E. Mezzetti, Linear congruences and hyperbolic systems of conservation laws, Projective varieties with unexpected properties, Walter de Gruyter GmbH & Co. KG, Berlin, 2005, pp. 209–230.
P. De Poi and E. Mezzetti, Congruences of lines in ℙ5, quadratic normality, and completely exceptional Monge–Ampère equations, Geom. Dedicata 131 (2008), no. 1, 213–230.
Ph. Ellia, Chern classes of rank two globally generated vector bundles on ℙ2, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24 (2013), no. 2, 147–163.
G. Fano, Reti di complessi lineari dello spazio S 5 aventi una rigata assegnata di rette-centri, Rend. Reale Accad. Naz. Lincei 11 (1930), 227–232.
D. Faenzi and M.L. Fania, Skew–symmetric matrices and Palatini scrolls, Math. Ann. 347 (2010), no. 4, 859–883.
M.L. Fania and E. Mezzetti, On the Hilbert scheme of Palatini threefolds, Adv. Geom. 2 (2002), no. 4, 371–389.
M.L. Fania and E. Mezzetti, Vector spaces of skew-symmetric matrices of constant rank, Linear Algebra Appl. 434 (2011), 2383–2403.
F.R. Gantmacher, The Theory of Matrices, vol. 2, Chelsea Publishing Company, New York, 1959.
B. Ilic and J.M. Landsberg, On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual varieties, Math. Ann. 314 (1999), no. 1, 159–174.
L. Manivel and E. Mezzetti, On linear spaces of skew-symmetric matrices of constant rank, Manuscripta Math. 117 (2005), no. 3, 319–331.
G. Ottaviani, On 3–folds in ℙ5 which are scrolls, Ann. Scuola Norm. Sup. Pisa Cl. Sc. (4) 19 (1992), no. 3, 451–471.
F. Palatini, Sui sistemi lineari dei complessi lineari di rette nello spazio a cinque dimensioni, Atti del R. Ist. Veneto di Scienze, Lettere e Arti 60 (1900), 371–383.
M. Schneider, Vector bundles and low codimensional submanifolds of projective space: a problem list, Topics in algebra, Part 2 (PWN, ed.), vol. 26, Banach Center Publ., 1990, pp. 209–222.
C. Segre, Studio sulle quadriche in uno spazio lineare ad un numero qualunque di dimensioni, Mem. R. Acc. Sc. Torino (1883a), 3–86.
C. Segre, Sulla geometria della retta e delle sue serie quadratiche, Mem. R. Acc. Sc. Torino (1883b), 87–157.
C. Segre, Un’osservazione sui sistemi di rette degli spazi superiori, Rend. Circ. Mat. Palermo (1888), 167–168.
C. Segre, Preliminari di una teoria delle varietà luoghi di spazi, Rend. Circ. Mat. Palermo (1910), 87–121.
C. Segre, Sui complessi lineari di piani nello spazio a cinque dimensioni, Ann. Mat. Pura Appl. (1917), 75–123.
C. Segre, Sulla varietà cubica con dieci punti doppi dello spazio a quattro dimensioni, Atti R. Acc. Scienze Torino 22 (1886–87), 791–801.
J.C. Sierra and L. Ugaglia, On double Veronese embeddings in the Grassmannian ℙ(1,N), Math. Nachr. 279 (2006), no. 7, 798–804.
J. Sylvester, On the dimension of spaces of linear transformations satisfying rank conditions, Linear Algebra Appl. 78 (1986), 1–10.
F. Tanturri, Degeneracy loci of morphisms between vector bundles, Ph.D. thesis, SISSA, Trieste, 2013.
F. Tanturri, Degeneracy loci of twisted differential forms and linear line complexes, Arch. Math. (Basel) 105 (2015), no. 2, 109–118.
F. Tanturri, On the Hilbert scheme of degeneracy loci of twisted differential forms, Trans. Amer. Math. Soc. 368 (2016), no. 7, 4561–4583.
R. Westwick, Spaces of matrices of fixed rank, Linear and Multilinear Algebra 20 (1987), no. 2, 171–174.
R. Westwick, Spaces of matrices of fixed rank. II, Linear Algebra Appl. 235 (1996), 163–169.
Acknowledgements
Emilia Mezzetti is member of INdAM – GNSAGA, and is supported by PRIN “Geometry of algebraic varieties” and by FRA, Fondi di Ricerca di Ateneo, University of Trieste.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this paper
Cite this paper
Mezzetti, E. (2016). Geometry of Lines and Degeneracy Loci of Morphisms of Vector Bundles. In: Casnati, G., Conte, A., Gatto, L., Giacardi, L., Marchisio, M., Verra, A. (eds) From Classical to Modern Algebraic Geometry. Trends in the History of Science. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-32994-9_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-32994-9_13
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-32992-5
Online ISBN: 978-3-319-32994-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)