Seshadri constants and Grassmann bundles over curves

Biswas, Indranil; Hanumanthu, Krishna; Nagaraj, Donihakkalu Shankar; Newstead, Peter E.

doi:10.5802/aif.3370

Biswas, Indranil ¹; Hanumanthu, Krishna ²; Nagaraj, Donihakkalu Shankar ³; Newstead, Peter E.⁴

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005 (India)
² Chennai Mathematical Institute, H1 SIPCOT IT Park, Siruseri, Kelambakkam 603103 (India)
³ The Institute of Mathematical Sciences (HBNI), CIT Campus, Taramani, Chennai 600113 (India); Indian Institute of Science Education and Research, Tirupati, Rami Reddy Nagar, Karakambadi Road, Mangalam (P.O.) Tirupati 517507 (India)
⁴ Department of Mathematical Sciences, University of Liverpool, Peach Street, Liverpool, L69 7ZL (England)

Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1477-1496.

Abstract
Résumé

Let $X$ be a smooth complex projective curve, and let $E$ be a vector bundle on $X$ which is not semistable. For a suitably chosen integer $r$ , let $Gr (E)$ be the Grassmann bundle over $X$ that parametrizes the quotients of the fibers of $E$ of dimension $r$ . Assuming some numerical conditions on the Harder–Narasimhan filtration of $E$ , we study Seshadri constants of ample line bundles on $Gr (E)$ . In many cases, we give the precise values of the Seshadri constants. Our results generalize various known results for the special case of $rank (E) = 2$ . We include some examples in rank $4$ .

Soient $X$ une courbe projective complexe lisse et $E$ un fibré vectoriel sur $X$ , supposé non semi-stable. Pour un entier $r$ bien choisi, considérons $Gr (E)$ le fibré de Grassman au-dessus de $X$ qui paramètre les quotients de dimension $r$ des fibrés de $E$ . Sous des hypothèses numériques sur la filtration de Harder–Narasimhan de $E$ , nous étudions les constantes de Seshadri des fibrés amples au-dessus de $Gr (E)$ . Dans de nombreux cas, nous obtenons les valeurs explicites des constantes de Seshadri. Nos résultats généralisent plusieurs résultats connus dans le cas particulier où $rang (E) = 2$ . Nous présentons des exemples en rang $4$ .

Received: 2018-04-21
Revised: 2018-10-01
Accepted: 2019-05-21
Published online: 2021-04-15

DOI: 10.5802/aif.3370
Classification: 14H60, 14Q20, 14C20
Keywords: Seshadri constant, Harder–Narasimhan filtration, Grassmann bundle, nef cone, pseudo-effective cone
Author's affiliations:

Biswas, Indranil ¹; Hanumanthu, Krishna ²; Nagaraj, Donihakkalu Shankar ³; Newstead, Peter E. ⁴

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     title = {Seshadri constants and {Grassmann} bundles over curves},
     journal = {Annales de l'Institut Fourier},
     pages = {1477--1496},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Biswas, Indranil; Hanumanthu, Krishna; Nagaraj, Donihakkalu Shankar; Newstead, Peter E. Seshadri constants and Grassmann bundles over curves. Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1477-1496. doi : 10.5802/aif.3370. https://aif.centre-mersenne.org/articles/10.5802/aif.3370/

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