Seshadri constants and Grassmann bundles over curves
Annales de l'Institut Fourier, Volume 70 (2020) no. 4, pp. 1477-1496.

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.

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