Geodesic
Seshadri constants on $\mathbb {P}^1\times \mathbb {P}^1$ and applications to the symplectic packing problem
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e191

Voir la notice de l'article provenant de la source Cambridge University Press

In this paper we compute the r-point Seshadri constant on $\mathbb {P}^1\times \mathbb {P}^1$ for those line bundles where the answer might be expected to be governed by $(-1)$-curves. As a consequence we obtain explicit formulas for the symplectic packing problem for $\mathbb {P}^1\times \mathbb {P}^1$. Some exact values for the r-point Seshadri constant outside the region governed by Mori’s cone theorem are also given. These latter results use a useful new “reflection method”.In the analysis there is a striking difference between the cases when r is odd and when r is even. When r is even the problem admits an infinite order automorphism, and there are infinitely many $(-1)$-curves to consider. In contrast, when r is odd only a finite number (usually four) types of $(-1)$-curves are relevant to our answer. 
Dionne, Chris; Roth, Mike. Seshadri constants on $\mathbb {P}^1\times \mathbb {P}^1$ and applications to the symplectic packing problem. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e191. doi: 10.1017/fms.2025.10137
@article{10_1017_fms_2025_10137,
     author = {Dionne, Chris and Roth, Mike},
     title = {Seshadri constants on $\mathbb {P}^1\times \mathbb {P}^1$ and applications to the symplectic packing problem},
     journal = {Forum of Mathematics, Sigma},
     pages = {e191},
     year = {2025},
     volume = {13},
     number = {1},
     doi = {10.1017/fms.2025.10137},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10137/}
}
TY  - JOUR
AU  - Dionne, Chris
AU  - Roth, Mike
TI  - Seshadri constants on $\mathbb {P}^1\times \mathbb {P}^1$ and applications to the symplectic packing problem
JO  - Forum of Mathematics, Sigma
PY  - 2025
SP  - e191
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10137/
DO  - 10.1017/fms.2025.10137
ID  - 10_1017_fms_2025_10137
ER  - 
%0 Journal Article
%A Dionne, Chris
%A Roth, Mike
%T Seshadri constants on $\mathbb {P}^1\times \mathbb {P}^1$ and applications to the symplectic packing problem
%J Forum of Mathematics, Sigma
%D 2025
%P e191
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/fms.2025.10137/
%R 10.1017/fms.2025.10137
%F 10_1017_fms_2025_10137

[1] Biran, P., ‘Symplectic packing in dimension 4’, Geom. Funct. Anal. 7(3) (1997), 420–437.10.1007/s000390050014 Google Scholar | DOI

[2] Biran, P., ‘From symplectic packing to algebraic geometry and back’, in European Congress of Mathematics, Vol. II (Barcelona, 2000), 507–524, Progr. Math., 202, Birkhäuser, Basel, 2001.10.1007/978-3-0348-8266-8_44 Google Scholar | DOI

[3] Ciliberto, C., Harbourne, B., Miranda, R., and Roé, J., ‘Variations of Nagata’s conjecture’, in A celebration of algebraic geometry, (Clay Math. Proc., 18 American Mathematical Society, Providence, RI, 2013), 185–203. Google Scholar

[4] Dionne, C., Numerical Restrictions on Seshadri Curves, with applications to , Ph.D. thesis, (Queen’s University, 2021). Google Scholar

[5] De Volder, C. and Tutaj-Gasińska, H., ‘Bound on Seshadri constants on . II’, Bull. Belg. Math. Soc. Simon Stevin 16(3) (2009), 463–468. Google Scholar

[6] Gromov, M., ‘Pseudo-holomorphic curves in symplectic manifolds’, Invent. Math. 82(2) (1985), 307–345.10.1007/BF01388806 Google Scholar | DOI

[7] Küchle, O., ‘Multiple point Seshadri constants and the dimension of adjoint linear series’, Ann. Inst. Fourier, 46 (1996), 63–71.10.5802/aif.1506 Google Scholar | DOI

[8] Lalonde, F. and Mcduff, D., ‘The classification of ruled symplectic 4-manifolds’, Math. Res. Lett. 3(6) (1996), 769–778.10.4310/MRL.1996.v3.n6.a5 Google Scholar | DOI

[9] Manin, Yu. I., Cubic Forms, Algebra, geometry, arithmetic. Translated from the Russian by M. Hazewinkel. Second edition (North-Holland Math. Library, 4, North-Holland Publishing Co., 1986). Google Scholar

[10] Mcduff, D., and Polterovich, L., ‘Symplectic packings and algebraic geometry’, Invent. Math. 115(3) (1994), 405–434.10.1007/BF01231766 Google Scholar | DOI

[11] Petrakiev, I., ‘Homogeneous interpolation and some continued fractions’, Trans. Amer. Math. Soc. Ser. B, 1 (2014), 23–44.10.1090/S2330-0000-2014-00001-0 Google Scholar | DOI

[12] Syzdek, W., ‘Submaximal Riemann-Roch expected curves and symplectic packing’, Ann. Acad. Pedagog. Crac. Stud. Math. 6 (2007), 101–122. Google Scholar

Cité par Sources :