We compute a class of Sarkisov links from Fano 3-folds embedded in weighted Grassmannians using explicit methods for describing graded rings associated to a variation of geometric invariant theory (GIT) quotient.

Let $X$ be an arithmetic surface, and let $L$ be a line bundle on $X$. Choose a metric $h$ on the lattice $\Lambda$ of sections of $L$ over $X$. When the degree of the generic fiber of $X$ is large enough, we get lower bounds for the successive minima of $(\Lambda ,h)$ in terms of the normalized height of $X$. The proof uses an effective version (due to C. Voisin) of a theorem of Segre on linear projections and Morrison's proof that smooth projective curves of high degree are Chow semistable.

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types $A$ and $C$, and the direct method for types $A$, $D$, and $B$ (level 2).

The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite series $G(\mathit{\text{de}},e,r)$, thus completing their calculation for all complex reflection groups.