On the Hilbert scheme of degeneracy loci of twisted differential forms

Tanturri, Fabio

doi:10.1090/tran/6637

On the Hilbert scheme of degeneracy loci of twisted differential forms
HTML articles powered by AMS MathViewer

by Fabio Tanturri PDF

Trans. Amer. Math. Soc. 368 (2016), 4561-4583 Request permission

Abstract:

We prove that, for $3 < m < n-1$, the Grassmannian of $m$-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension $n$ is birational to the Hilbert scheme of degeneracy loci of $m$ global sections of $\Omega _{\mathbb {P}^{n-1}}(2)$, the twisted cotangent bundle on $\mathbb {P}^{n-1}$. For $3=m<n-1$ and $n$ odd, this Grassmannian is proved to be birational to the set of Veronese surfaces parameterized by the Pfaffians of linear skew-symmetric matrices of order $n$.

References

Constantin Bănică, Smooth reflexive sheaves, Proceedings of the Colloquium on Complex Analysis and the Sixth Romanian-Finnish Seminar, 1991, pp. 571–593. MR 1172165
W. Barth, Some properties of stable rank-$2$ vector bundles on $\textbf {P}_{n}$, Math. Ann. 226 (1977), no. 2, 125–150. MR 429896, DOI 10.1007/BF01360864
David A. Buchsbaum and David Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension $3$, Amer. J. Math. 99 (1977), no. 3, 447–485. MR 453723, DOI 10.2307/2373926
Dolores Bazan and Emilia Mezzetti, On the construction of some Buchsbaum varieties and the Hilbert scheme of elliptic scrolls in $\Bbb P^5$, Geom. Dedicata 86 (2001), no. 1-3, 191–204. MR 1856426, DOI 10.1023/A:1011926128564
Guido Castelnuovo, Ricerche di geometria della retta nello spazio a quattro dimensioni, Atti R. Ist. Veneto Sc., 2:855–901, 1891.
David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. MR 1730819
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Gino Fano, Reti di complessi lineari dello spazio ${S}_5$ aventi una rigata assegnata di rette-centri, Rend. Reale Accad. Naz. Lincei, XI:227–232, 1930.
Daniele Faenzi and Maria Lucia Fania, Skew-symmetric matrices and Palatini scrolls, Math. Ann. 347 (2010), no. 4, 859–883. MR 2658146, DOI 10.1007/s00208-009-0450-5
Daniele Faenzi and Maria Lucia Fania, On the Hilbert scheme of varieties defined by maximal minors, Math. Res. Lett. 21 (2014), no. 2, 297–311. MR 3247058, DOI 10.4310/MRL.2014.v21.n2.a8
Maria Lucia Fania and Emilia Mezzetti, On the Hilbert scheme of Palatini threefolds, Adv. Geom. 2 (2002), no. 4, 371–389. MR 1940444, DOI 10.1515/advg.2002.017
M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Mathematics, Vol. 14, Springer-Verlag, New York-Heidelberg, 1973. MR 0341518
Laurent Gruson and Christian Peskine, Courbes de l’espace projectif: variétés de sécantes, Enumerative geometry and classical algebraic geometry (Nice, 1981), Progr. Math., vol. 24, Birkhäuser, Boston, Mass., 1982, pp. 1–31 (French). MR 685761
Alexander Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.], Secrétariat mathématique, Paris, 1962 (French). MR 0146040
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Robin Hartshorne, Stable reflexive sheaves, Math. Ann. 254 (1980), no. 2, 121–176. MR 597077, DOI 10.1007/BF01467074
Robin Hartshorne, Deformation theory, Graduate Texts in Mathematics, vol. 257, Springer, New York, 2010. MR 2583634, DOI 10.1007/978-1-4419-1596-2
F. S. Macaulay, The algebraic theory of modular systems, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1994. Revised reprint of the 1916 original; With an introduction by Paul Roberts. MR 1281612
Shigeru Mukai, Fano $3$-folds, Complex projective geometry (Trieste, 1989/Bergen, 1989) London Math. Soc. Lecture Note Ser., vol. 179, Cambridge Univ. Press, Cambridge, 1992, pp. 255–263. MR 1201387, DOI 10.1017/CBO9780511662652.018
Shigeru Mukai, Curves and symmetric spaces. I, Amer. J. Math. 117 (1995), no. 6, 1627–1644. MR 1363081, DOI 10.2307/2375032
Giorgio Ottaviani, On $3$-folds in $\mathbf P^5$ which are scrolls, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 19 (1992), no. 3, 451–471. MR 1205407
Francesco Palatini, Sui sistemi lineari di complessi lineari di rette nello spazio a cinque dimensioni, Atti R. Ist. Veneto Sc., (60):371–383, 1901.
Francesco Palatini, Sui complessi lineari di rette negli iperspazi, Giorn. di Mat., (41):85–96, 1903.
Frank-Olaf Schreyer, Geometry and algebra of prime Fano 3-folds of genus 12, Compositio Math. 127 (2001), no. 3, 297–319. MR 1845040, DOI 10.1023/A:1017529016445
Fabio Tanturri, Pfaffian representations of cubic surfaces, Geom. Dedicata 168 (2014), 69–86. MR 3158031, DOI 10.1007/s10711-012-9818-x
Fabio Tanturri, Degeneracy loci of twisted differential forms and linear line complexes, Arch. Math. (Basel) 105 (2015), no. 2, 109–118. MR 3372594, DOI 10.1007/s00013-015-0768-z
Jerzy Weyman, Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, vol. 149, Cambridge University Press, Cambridge, 2003. MR 1988690, DOI 10.1017/CBO9780511546556