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In this paper we compute the r -point Seshadri constant on$\\mathbb {P}^1\\times \\mathbb {P}^1$for those line bundles where the answer might be expected to be governed by$(-1)$-curves. As a consequence we obtain explicit formulas for the symplectic packing problem for$\\mathbb {P}^1\\times \\mathbb {P}^1$. Some exact values for the r -point Seshadri constant outside the region governed by Mori’s cone theorem are also given. These latter results use a useful new “reflection method”. In the analysis there is a striking difference between the cases when r is odd and when r is even. When r is even the problem admits an infinite order automorphism, and there are infinitely many$(-1)$-curves to consider. In contrast, when r is odd only a finite number (usually four) types of$(-1)$-curves are relevant to our answer.

Let$p$be an odd prime. We construct a$p$-group$P$of nilpotency class two, rank seven and exponent$p$, such that$\\text{Aut}(P)$induces$N_{\\text{GL}(7,p)}(G_{2}(p))=Z(\\text{GL}(7,p))G_{2}(p)$on the Frattini quotient$P/\\unicode[STIX]{x1D6F7}(P)$. The constructed group$P$is the smallest$p$-group with these properties, having order$p^{14}$, and when$p=3$our construction gives two nonisomorphic$p$-groups. To show that$P$satisfies the specified properties, we study the action of$G_{2}(q)$on the octonion algebra over$\\mathbb{F}_{q}$, for each power$q$of$p$, and explore the reducibility of the exterior square of each irreducible seven-dimensional$\\mathbb{F}_{q}[G_{2}(q)]$-module.

We prove transitivity for volume-preserving [formula omitted: see PDF] diffeomorphisms on [formula omitted: see PDF] which are isotopic to a linear Anosov automorphism along a path of weakly partially hyperbolic diffeomorphisms.

For a finite group

In this paper, we prove the equidistribution of saddle periodic points for Hénon-type automorphisms of C k with respect to its equilibrium measure. A general strategy to obtain equidistribution properties in any dimension is presented. It is based on our recent theory of densities for positive closed currents. Several fine properties of dynamical currents are also proved.

For the queer Lie superalgebra Q (1), we explicitly describe the center of the super-Yangian Y Q (1) of Q (1) in terms of generators of Y Q (1). We also describe the square of the antipodal map on Y Q (1) and use the antipode to construct an involutive automorphism of Y Q (1).

The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism If the cancellation does not hold then

It is shown that if$G$is a finite$p$-group of coclass 2 with$p>2$, then$G$has a noninner automorphism of order$p$.

In this paper we prove that surfaces of general type with irregularity$q\\geq 3$are rationally cohomologically rigidified, and so are minimal surfaces$S$with$q(S)= 2$unless${ K}_{S}^{2} = 8\\chi ({ \\mathcal{O} }_{S} )$. Here a surface$S$is said to be rationally cohomologically rigidified if its automorphism group$\\mathrm{Aut} (S)$acts faithfully on the cohomology ring${H}^{\\ast } (S, \\mathbb{Q} )$. As examples we give a complete classification of surfaces isogenous to a product with$q(S)= 2$that are not rationally cohomologically rigidified.