Let $\Delta$ be an Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of $\Delta$ are normal and Cohen--Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

We show that every finite, finitely related algebra in a congruence distributive variety has a near unanimity term operation. As a consequence we solve the near unanimity problem for relational structures: it is decidable whether a given finite set of relations on a finite set admits a compatible near unanimity operation. This consequence also implies that it is decidable whether a given finite constraint language defines a constraint satisfaction problem of bounded strict width.

In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold that is modular, in the sense that the $L$-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55.

We study the regularity of the higher secant varieties of $\PP^1\times \PP^n$, embedded with divisors of type $(d,2)$ and $(d,3)$. We produce, for the highest defective cases, a ``determinantal'' equation of the secant variety. As a corollary, we prove that the Veronese triple embedding of $\PP^n$ is not Grassmann defective.

K.~Ding studied a class of Schubert varieties $X_\lambda$ in type A partial flag manifolds, indexed by integer partitions $\lambda$ and in bijection with dominant permutations. He observed that the Schubert cell structure of $X_\lambda$ is indexed by maximal rook placements on the Ferrers board $B_\lambda$, and that the integral cohomology groups $H^*(X_\lambda;\:\Zz)$, $H^*(X_\mu;\:\Zz)$ are additively isomorphic exactly when the Ferrers boards $B_\lambda, B_\mu$ satisfy the combinatorial condition of \emph{rook-equivalence}. We classify the varieties $X_\lambda$ up to isomorphism, distinguishing them by their graded cohomology rings with integer coefficients. The crux of our approach is studying the nilpotence orders of linear forms in the cohomology ring.

We study $K$-orbits in $G/P$ where $G$ is a complex connected reductive group, $P \subseteq G$ is a parabolic subgroup, and $K \subseteq G$ is the fixed point subgroup of an involutive automorphism $\theta$. Generalizing work of Springer, we parametrize the (finite) orbit set $K \setminus G \slash P$ and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of $\theta$-stable (resp. $\theta$-split) parabolic subgroups. We also describe the decomposition of any $(K,P)$-double coset in $G$ into $(K,B)$-double cosets, where $B \subseteq P$ is a Borel subgroup. Finally, for certain $K$-orbit closures $X \subseteq G/B$, and for any homogeneous line bundle $\mathcal{L}$ on $G/B$ having nonzero global sections, we show that the restriction map $\res_X \colon H^0 (G/B, \mathcal{L}) \to H^0 (X, \mathcal{L})$ is surjective and that $H^i (X, \mathcal{L}) = 0$ for $i \geq 1$. Moreover, we describe the $K$-module $H^0 (X, \mathcal{L})$. This gives information on the restriction to $K$ of the simple $G$-module $H^0 (G/B, \mathcal{L})$. Our construction is a geometric analogue of Vogan and Sepanski's approach to extremal $K$-types.

For an arbitrary finite Galois $p$-extension $L/K$ of $\zp$-cyclotomic number fields of $\CM$-type with Galois group $G = \Gal(L/K)$ such that the Iwasawa invariants $\mu_K^-$, $ \mu_L^-$ are zero, we obtain unconditionally and explicitly the Galois module structure of $\clases$, the minus part of the $p$-subgroup of the class group of $L$. For an arbitrary finite Galois $p$-extension $L/K$ of algebraic function fields of one variable over an algebraically closed field $k$ of characteristic $p$ as its exact field of constants with Galois group $G = \Gal(L/K)$ we obtain unconditionally and explicitly the Galois module structure of the $p$-torsion part of the Jacobian variety $J_L(p)$ associated to $L/k$.

We make precise the structure of the first two reduction morphisms associated with codimension two non-singular subvarieties of non-singular quadrics $\Q^n$, $n\geq 5$. We give a coarse classification of the same class of subvarieties when they are assumed not to be of log-general-type.}