What is it known now about Nagata's conjecture and Seshadri constant (http://en.wikipedia.org/wiki/Nagata%27s_conjecture_on_curves and http://en.wikipedia.org/wiki/Seshadri_constant) for toric surfaces? It seems that it should be some lower bounds in terms of fans or polytops. Is it true?

Does there exist some simple examples where exact inequalities are proved?

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Hi Nikita, I don't know much about Nagata's Conjecture, but Seshadri constants can be calculated directly for (smooth, complete) toric surfaces (at the distinguished toric fixed points). Is this what you are wondering about? –  Robert Garbary Dec 28 '11 at 20:07
Robert, as I understand we can calculate it only for fixed points (but yes, I was wondering about answer in this case too), so there is question how to calculate inf $\frac{C\cdot L}{mult_x C}$ for general point $x$ in other toric orbits. –  Nikita Kalinin Dec 28 '11 at 20:33

I assume $S$ is a projective smooth toric surface.

If $D_1, \dots, D_n$ are the irreducible toric divisors on $S$, then $-K_S=D_1+\dots+D_n$ is an anticanonical divisor. Thus blowing up any point on any of these divisors one obtains a smooth anticanonical rational surface $\tilde S$; such surfaces are very well known by work of Brian Harbourne. Maybe more simply, by Mori theory, since $-K_{\tilde S}=-\pi^*K_S-E$ is effective on $\tilde S$, the cone of curves is spanned by extremal rays $\mathbb{R}C_i$ with $-K_{\tilde S}\cdot C_i>0$ and components of $-K$; these curves are in our case the exceptional divisor and the birational transforms of a subset of $D_i$'s [EDIT: the previous (italicized) sentence was not correct because new extremal rays do appear with the blowup, but it is still true that these extremal rays can be controlled with Harbourne's results]. Thus to determine nefness on $\tilde S$ and so the Seshadri constant of any ample divisor at the given point is not difficult. In particular these Seshadri constants are rational, and they only depend on the ample class and the $D_i$(s) to which the point belongs. [EDIT: these conclusions are still correct].

Blowing up at a general point of $S$ may give a surface which is anticanonical (if $-K$ is not fixed on $S$) or non-anticanonical (when $|-K|=\{-K\}$). In the first case, similar considerations would lead to the computation of the Seshadri constant. In the second case, I am afraid the problem can be difficult, and indeed related to the Nagata conjecture. The simplest interesting example would be the following: start with $\mathbb{P}^2$ as a toric surface and blow it up at the three three toric points. Now blow up the resulting surface at its six toric points. The result is a toric surface, the blow up of $\mathbb{P}^2$ at three clusters of three infinitely near points, where the Seshadri constant of your preferred ample divisor $L$ at a general point is presumably unknown. It might be irrational, if $L^2$ is not a square.

An analogon of the Nagata conjecture for toric surfaces (Seshadri constants at sets of $r\gg 0$ general points) can of course be stated, as particular cases of the conjecture stated by Lazarsfeld in 5.1.24 of "Positivity in Algebraic Geometry". I have nothing particularly relevant to say about that, except that it will probably be just as difficult as Nagata's original conjecture.

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Atsushi Ito recently uploaded a relevant preprint: arXiv:1202.6664. I had no time to go through it, but the abstract says: "We give lower and upper bounds of Seshadri constants on toric varieties at any points. By using the lower bounds and toric degenerations, we can obtain some new computations or estimations of Seshadri constants on non-toric varieties. In particular, we investigate Seshadri constants on hypersurfaces in projective spaces and Fano 3-folds with Picard number one in detail." –  quim Mar 2 '12 at 10:46

On toric varieties, I think it is more convenient to work with the following definition of Seshadri. Let $p \in X$ and let $D$ be a $NEF$ divisor on $X$. Let $\pi : \hat{X} \to X$ denote the blow-up of $X$ at $p$, with $E$ being the exceptional divisor. Then the Seshadri constant of $(X,p,D)$ is defined to be the largest $r \in \mathbb{R}$ so that $\pi^* D - rE$ is NEF (on $\hat{X}$). (This definition is 'true' for all varieties, there is nothing special about toric-ness here.)

It's more convenient because, on toric varieties, we have a very nice description of blowing up at the $T$-invariant points - just insert an extra ray into the fan. On a (smooth, complete? I always forget which assumptions are needed) toric variety $X$, a divisor $D$ is NEF iff $D.C \geq 0$ for all $T$-invariant complete curves on the variety. Thus, this turns an infinite problem (calculating $D.C$ for every curve $C \subseteq X$) into a finite problem (calculating $D.C$ for every $C$ that corresponds to a ray of the fan). Since we know how to compute $D.C$, you can explicitly compute the $s$ mentioned in the above definition. This only works for the $T$-invariant points.

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Yes, sure. But what is it known about inf $D.C /...$ on other stratums? –  Nikita Kalinin Jan 9 '12 at 20:09