Towards the Bertram-Feinberg-Mukai Conjecture

In this paper, we prove the existence portion of the Bertram-Feinberg-Mukai Conjecture for an infinite family of new cases using degeneration technique. This not only leads to a substantial improvement of known results but also develops finer tools for analyzing the moduli of rank two limit linear series which should be useful for other applications to other higher-rank Brill-Noether Problems

The variety of reductions for a reductive symmetric pair

We define and study the variety of reductions for a reductive symmetric pair (G,theta), which is the natural compactification of the set of the Cartan subspaces of the symmetric pair. These varieties generalize the varieties of reductions for the Severi varieties studied by Iliev and Manivel, which are Fano varieties. We develop a theoretical basis to the study these varieties of reductions, and relate the geometry of these variety to some problems in representation theory. A very useful result is the rigidity of semi-simple elements in deformations of algebraic subalgebras of Lie algebras. We apply this theory to the study of other varieties of reductions in a companion paper, which yields two new Fano varieties.Comment: 23 page

Functional renormalization group approach to non-collinear magnets

A functional renormalization group approach to $d$-dimensional, $N$-component, non-collinear magnets is performed using various truncations of the effective action relevant to study their long distance behavior. With help of these truncations we study the existence of a stable fixed point for dimensions between $d= 2.8$ and $d=4$ for various values of $N$ focusing on the critical value $N_c(d)$ that, for a given dimension $d$, separates a first order region for $NN_c(d)$. Our approach concludes to the absence of stable fixed point in the physical - $N=2,3$ and $d=3$ - cases, in agreement with $\epsilon=4-d$-expansion and in contradiction with previous perturbative approaches performed at fixed dimension and with recent approaches based on conformal bootstrap program.Comment: 16 pages, 8 figure