Cox rings of surfaces with nef anticanonical class
Michela Artebani
Let $X$ be a smooth complex projective rational surface with $q(X)=0$ and nef anticanonical divisor $-K_X$.
The Cox ring [1]
$R(X)=\bigoplus_{[D]\in \mathrm{Cl}(X)} \Gamma(X,\mathcal O_X(D))$
encodes the effective divisor theory of $X$.
The ring is known to be finitely generated when $-K_X$ is big [4,6]
or when $\kappa(-K_X)=1$ and the effective cone is polyhedral [3].
In this talk we present a uniform and explicit description of generators for $R(X)$
that depends only on the anticanonical geometry of $X$ (the linear system $|-K_X|$ and the configuration of negative curves).
Our construction extends the known descriptions for generalized del Pezzo surfaces [4,5]
and for extremal rational elliptic surfaces [2].
This is joint work with Sofía Pérez Garbayo [7].
References:
$[1]$ I. Arzhantsev, U. Derenthal, J. Hausen, A. Laface, Cox rings, Cambridge Studies in Advanced
Mathematics, vol. 144, Cambridge University Press, Cambridge, 2015
$[2]$ M. Artebani, A. Garbagnati, A. Laface, Cox rings of extremal rational elliptic surfaces, Trans.
Amer. Math. Soc. 368 (2016), no. 3, 1735–1757.
$[3]$ M. Artebani and A. Laface, Cox rings of surfaces and the anticanonical Iitaka dimension,
Adv. Math. 226 (2011), no. 6, 5252–5267, DOI 10.1016/j.aim.2011.01.007.
$[4]$ V.V. Batyrev and O.N. Popov, The Cox ring of a del Pezzo surface, Arithmetic of higher-dimensional algebraic varieties (Palo Alto, CA, 2002), 2004, pp. 85–103.
$[5]$ U. Derenthal, Geometrie universeller Torsore, Doctoral thesis, Fakultät für Mathematik und
Informatik, Georg August Universität Göttingen, 2006.
$[6]$ D. Testa, A. V\'arilly-Alvarado, M. Velasco, Big rational surfaces, Math. Ann. 351 (2011),no. 1, 95–107.
$[7]$ M. Artebani, S. P\'erez Garbayo, Cox rings of nef anticanonical rational surfaces, Adv. Math. 475 (2025).