Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras $P(n)$, $n\ge 2$, and on the simple associative superalgebras $M(m,n)$, $m,n\ge 1$, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra $A(m,n)$ that are induced from G-gradings on $M(m+1,n+1)$. In the case of Lie superalgebras, the characteristic is assumed to be 0.     

The spectrum of resolvable holey Mendelsohn triple systems and holey Mendelsohn frames with block size three

A holey Mendelsohn triple system (HMTS) is a decomposition of a complete multipartite directed graph into directed cycles of length 3. If the directed cycles of length 3 can be partitioned into parallel classes, then the HMTS is called an RHMTS. Bennett, Wei and Zhu [J. Combin. Des., 1997] showed that an RHMTS of type $g^n$ exists when $gn\equiv 0(\mathrm{mod}3)$ and $(g,n)\ne (1,6)$ with some possible exceptions. In this paper, motivated by the application in constructing RHMTSs, we investigate the constructions of holey Mendelsohn frames. We prove that a 3-MHF of type $(n,h^t)$ exists if and only if $n\ge 3$, $t\ge 4$ and $nh(t-1)\equiv 0(\mathrm{mod}3)$, and then determine that the necessary condition for the existence of an RHMTS of type $g^n$, namely, $gn\equiv 0(\mathrm{mod}3)$ is also sufficient except for $(g,n)=(1,6)$. New recursive constructions on incomplete RHMTSs via MHFs are introduced to settle this problem completely.     

The space of lines in cyclic covers of projective space

Take positive integers m, n and d. Let Y be an m-fold cyclic cover of ${\mathbb{P}}^n$ ramified over a general hypersurface $X\subset P^n$ of degree md. In this paper we study the space $F(Y)$ of lines in Y and show that it is smooth of dimension $2(n-1)-d(m-1)$ if $md>2n-3$ and $2(n-1)-d(m-1)\ge 0$. When $2(n-1)=d(m-1)$, our result gives a formula on the number of m-contact order lines of X (see Definition 1.2).