The minimal resolution of the degree four cyclic cover of the plane branched along a GIT stable quartic is a $K3$ surface with a non symplectic action of $\Z_4$. In this paper we study the geometry of the one-dimensional family of $K3$ surfaces associated to the locus of plane quartics with five nodes.

Let $\R$ be a real closed field, let $X \subset \R^n$ be an irreducible real algebraic set and let $Z$ be an algebraic subset of $X$ of codimension $\geq 2$. Dubois and Efroymson proved the existence of an irreducible algebraic subset of $X$ of codimension $1$ containing~$Z$. We improve this dimension theorem as follows. Indicate by $\mu$ the minimum integer such that the ideal of polynomials in $\R[x_1,\ldots,x_n]$ vanishing on $Z$ can be generated by polynomials of degree $\leq \mu$. We prove the following two results: \begin{inparaenum}[\rm(1)] \item There exists a polynomial $P \in \R[x_1,\ldots,x_n]$ of degree~$\leq \mu+1$ such that $X \cap P^{-1}(0)$ is an irreducible algebraic subset of $X$ of codimension $1$ containing~$Z$. \item Let $F$ be a polynomial in $\R[x_1,\ldots,x_n]$ of degree~$d$ vanishing on $Z$. Suppose there exists a nonsingular point $x$ of $X$ such that $F(x)=0$ and the differential at $x$ of the restriction of $F$ to $X$ is nonzero. Then there exists a polynomial $G \in \R[x_1,\ldots,x_n]$ of degree $\leq \max\{d,\mu+1\}$ such that, for each $t \in (-1,1) \setminus \{0\}$, the set $\{x \in X \mid F(x)+tG(x)=0\}$ is an irreducible algebraic subset of $X$ of codimension $1$ containing~$Z$. \end{inparaenum} Result (1) and a slightly different version of result~(2) are valid over any algebraically closed field also.

The ({\em classical}, {\em small quantum}, {\em equivariant}) cohomology ring of the grassmannian $G(k,n)$ is generated by certain derivations operating on an exterior algebra of a free module of rank $n$ ( Schubert calculus on a Grassmann algebra). Our main result gives, in a unified way, a presentation of all such cohomology rings in terms of generators and relations. Using results of Laksov and Thorup, it also provides a presentation of the universal factorization algebra of a monic polynomial of degree $n$ into the product of two monic polynomials, one of degree $k$.

In \emph{Connections on a parabolic principal bundle over a curve, I} we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in the above article that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalizes to the context of parabolic principal bundles. For usual principal bundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyah exact sequence.

A new tautological relation of $\overline{\mathcal{M}}_{3,1}$ in codimension 3 is derived and proved, using an invariance constraint from our previous work.

We present a new approach to noncommutative real algebraic geometry based on the representation theory of $C^\ast$-algebras. An important result in commutative real algebraic geometry is Jacobi's representation theorem for archimedean quadratic modules on commutative rings. We show that this theorem is a consequence of the Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras. A noncommutative version of Gelfand--Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's theorem to associative rings with involution.

Let $q = 2,3$ and $f(X,Y)$, $g(X,Y)$, $h(X)$ be polynomials with integer coefficients. In this paper we deal with the curve $f(X,Y)^q = h(X)g(X,Y)$, and we show that under some favourable conditions it is possible to determine all of its rational points.

In this paper we prove that the cone of effective divisors on the Kontsevich moduli spaces of stable maps, $\Kgnb{0,0}(\PP^r,d)$, stabilize when $r \geq d$. We give a complete characterization of the effective divisors on $\Kgnb{0,0}(\PP^d,d)$. They are non-negative linear combinations of boundary divisors and the divisor of maps with degenerate image.

We prove that for a generic hypersurface in $\mathbb P^{2n+1}$ of degree at least $2+2/n$, the $n$-th Picard number is one. The proof is algebraic in nature and follows from certain coherent cohomology vanishing.

The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by L.~Schl\"afli in 1858. Using this, explicit examples of surfaces of every possible type are given.

Let $k$ be a field of characteristic not $2,3$. Let $G$ be an exceptional simple algebraic group over $k$ of type $\F$, $^1{\E_6}$ or $\E_7$ with trivial Tits algebras. Let $X$ be a projective $G$-homogeneous variety. If $G$ is of type $\E_7$, we assume in addition that the respective parabolic subgroup is of type $P_7$. The main result of the paper says that the degree map on the group of zero cycles of $X$ is injective.

We construct several examples of higher-dimensional Calabi--Yau manifolds and prove their modularity.

In this paper we show that any Frobenius split, smooth, projective threefold over a perfect field of characteristic $p>0$ is Hodge--Witt. This is proved by generalizing to the case of threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt. We also show that the second crystalline cohomology of any smooth, projective Frobenius split variety does not have any exotic torsion. In the last two sections we include some applications.

Let $X$ be a smooth complex projective curve of genus $g\geq 1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let $W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$ by $\xi^{-1}$. There is a rational map $D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$, where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable vector bundles on $X$ with trivial determinant. We prove that if $n=2$, then $D_{\xi}$ is everywhere defined and is injective.

Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$ defined over $\Q$ whose rational points classify the quadratic $\Q$-curves of degree $N$ realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case $N=5$.

Dans le papier Beyond Endoscopy une id\'ee pour entamer la fonctorialit\'e en utilisant la formule des traces a \'et\'e introduite. Maints probl\`emes, l'existence d'une limite convenable de la formule des traces, est eqquiss\'ee dans cette note informelle mais seulement pour $GL(2)$ et les corps des fonctions rationelles sur un corps fini et en ne pas resolvant bon nombre de questions.

We prove that the elliptic surface $y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric Mordell--Weil rank $15$. This completes a list of Kuwata, who gave explicit examples of elliptic $K3$-surfaces with geometric Mordell--Weil ranks $0,1,\dots, 14, 16, 17, 18$.

It is shown that the coniveau filtration on the cohomology of smooth projective varieties is preserved up to shift by pushforwards, pullbacks and products.

Starting with the Leibniz algebra defined by a $\varphi$-dialgebra, we construct examples of ``coquecigrues,'' in the sense of Loday, that is to say, manifolds whose tangent structure at a distinguished point coincides with that of the Leibniz algebra. We discuss some possible implications and generalizations of this construction.

We study natural $*$-valuations, $*$-places and graded $*$-rings associated with $*$-ordered rings. We prove that the natural $*$-valuation is always quasi-Ore and is even quasi-commutative (\emph{i.e.,} the corresponding graded $*$-ring is commutative), provided the ring contains an imaginary unit. Furthermore, it is proved that the graded $*$-ring is isomorphic to a twisted semigroup algebra. Our results are applied to answer a question of Cimpri\v c regarding $*$-orderability of quantum groups.

In this article we will show that there are infinitely many symmetric, integral $3 \times 3$ matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire N\'eron--Severi group of this surface and find all low degree curves on it.

In this paper we describe six pencils of $K3$-surfaces which have large Picard number ($\rho=19,20$) and each contains precisely five special fibers: four have A-D-E singularities and one is non-reduced. In particular, we characterize these surfaces as cyclic coverings of some $K3$-surfaces described in a recent paper by Barth and the author. In many cases, using 3-divisible sets, resp., 2-divisible sets, of rational curves and lattice theory, we describe explicitly the Picard lattices.

The purpose of this note is to show that the homologically trivial cycles contructed by Clemens and their generalisations due to Paranjape can be detected by the technique of spreading out. More precisely, we associate to these cycles (and the ambient varieties in which they live) certain families which arise naturally and which are defined over $\bbC$ and show that these cycles, along with their relations, can be detected in the singular cohomology of the total space of these families.

We give a short proof of Totaro's theorem that every$E_8$-torsor over a field $k$ becomes trivial over a finiteseparable extension of $k$of degree dividing $d(E_8)=2^63^25$.

This paper investigates the modularity of three non-rigid Calabi--Yau threefolds with bad reduction at 11. They are constructed as fibre products of rational elliptic surfaces, involving the modular elliptic surface of level 5. Their middle $\ell$-adic cohomology groups are shown to split into two-dimensional pieces, all but one of which can be interpreted in terms of elliptic curves. The remaining pieces are associated to newforms of weight 4 and level 22 or 55, respectively. For this purpose, we develop a method by Serre to compare the corresponding two-dimensional 2-adic Galois representations with uneven trace. Eventually this method is also applied to a self fibre product of the Hesse-pencil, relating it to a newform of weight 4 and level 27.