Equivariant K-stability under finite group action

We show that [Formula: see text]-equivariant K-semistability (respectively, [Formula: see text]-equivariant K-polystability) implies K-semistability (respectively, K-polystability) for log Fano pairs with klt singularities when [Formula: see text] is a finite group.

Shimura subvarieties in the Prym locus of ramified Galois coverings

We study Shimura (special) subvarieties in the moduli space $$A_{p,D}$$ A p , D of complex abelian varieties of dimension p and polarization type D. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $${{\mathbb {P}}}^1$$ P 1 . We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.

Polynomial Invariants of the Euclidean Group Action on Multiple Screws

<p>In this work, we examine the polynomial invariants of the special Euclidean group in three dimensions, SE(3), in its action on multiple screw systems. We look at the problem of finding generating sets for these invariant subalgebras, and also briefly describe the invariants for the standard actions on R^n of both SE(3) and SO(3). The problem of the screw system action is then approached using SAGBI basis techniques, which are used to find invariants for the translational subaction of SE(3), including a full basis in the one and two-screw cases. These are then compared to the known invariants of the rotational subaction. In the one and two-screw cases, we successfully derive a full basis for the SE(3) invariants, while in the three-screw case, we suggest some possible lines of approach.</p>

Polynomial Invariants of the Euclidean Group Action on Multiple Screws

<p>In this work, we examine the polynomial invariants of the special Euclidean group in three dimensions, SE(3), in its action on multiple screw systems. We look at the problem of finding generating sets for these invariant subalgebras, and also briefly describe the invariants for the standard actions on R^n of both SE(3) and SO(3). The problem of the screw system action is then approached using SAGBI basis techniques, which are used to find invariants for the translational subaction of SE(3), including a full basis in the one and two-screw cases. These are then compared to the known invariants of the rotational subaction. In the one and two-screw cases, we successfully derive a full basis for the SE(3) invariants, while in the three-screw case, we suggest some possible lines of approach.</p>

Shimura curves in the Prym loci of ramified double covers

We study Shimura curves of PEL type in the space of polarized abelian varieties [Formula: see text] generically contained in the ramified Prym locus. We generalize to ramified double covers, the construction done in [E. Colombo, P. Frediani, A. Ghigi and M. Penegini, Shimura curves in the Prym locus, Commun. Contemp. Math. 21(2) (2019) 1850009] in the unramified case and in the case of two ramification points. Namely, we construct families of double covers which are compatible with a fixed group action on the base curve. We only consider the case of one-dimensional families and where the quotient of the base curve by the group is [Formula: see text]. Using computer algebra we obtain 184 Shimura curves contained in the (ramified) Prym loci.

New dualities from old: generating geometric, Petrie, and Wilson dualities and trialities of ribbon graphs

We define a new ribbon group action on ribbon graphs that uses a semidirect product of a permutation group and the original ribbon group of Ellis-Monaghan and Moffatt to take (partial) twists and duals, or twuals, of ribbon graphs. A ribbon graph is a fixed point of this new ribbon group action if and only if it is isomorphic to one of its (partial) twuals. This extends the original ribbon group action, which only used the canonical identification of edges, to the more natural setting of self-twuality up to isomorphism. We then show that every ribbon graph has in its orbit an orientable embedded bouquet and prove that the (partial) twuality properties of these bouquets propagate through their orbits. Thus, we can determine (partial) twualities via these one vertex graphs, for which checking isomorphism reduces simply to checking dihedral group symmetries. Finally, we apply the new ribbon group action to generate all self-trial ribbon graphs on up to seven edges, in contrast with the few, large, very high-genus, self-trial regular maps found by Wilson, and by Jones and Poultin. We also show how the automorphism group of a ribbon graph yields self-dual, -petrial or –trial graphs in its orbit, and produce an infinite family of self-trial graphs that do not arise as covers or parallel connections of regular maps, thus answering a question of Jones and Poulton.