In this paper we study genus $2$ curves whose Jacobians admit a polarized $(4,4)$-isogeny to a product of elliptic curves. We consider base fields of characteristic different from $2$ and $3$, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their $4$-torsion, and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus $2$ curves, where previously only Richelot isogenies with kernels that are pointwise defined over the base field were considered. Our main tool is a Galois theoretic characterization of genus $2$ curves admitting multiple Richelot isogenies.

Generalized étale homotopy pro-groups $\pi_1^{\operatorname{ét}}(ċ{C}, x)$ associated with pointed, connected, small Grothendieck sites $(\mathcal{C}, x)$ are defined, and their relationship to Galois theory and the theory of pointed torsors for discrete groups is explained.
Applications include new rigorous proofs of some folklore results around $\pi_1^{\operatorname{ét}}(ét(X), x)$, a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new étale van Kampen theorem that gives a simple statement about a pushout square of pro-groups that works for covering families that do not necessarily consist exclusively of monomorphisms. A corresponding van Kampen result for Grothendieck's profinite groups $\pi_1^{\mathrm{Gal}}$ immediately follows.

The moduli space of smooth real binary octics has five connected components. They parametrize the real binary octics whose defining equations have $0,\dots,4$ complex-conjugate pairs of roots respectively. We show that each of these five components has a real hyperbolic structure in the sense that each is isomorphic as a real-analytic manifold to the quotient of an open dense subset of $5$-dimensional real hyperbolic space $\mathbb{RH}^5$ by the action of an arithmetic subgroup of $\operatorname{Isom}(\mathbb{RH}^5)$. These subgroups are commensurable to discrete hyperbolic reflection groups, and the Vinberg diagrams of the latter are computed.

Speyer and Sturmfels associated Gröbner toric degenerations $\mathrm{Gr}_2(\mathbb{C}^n)^{\mathcal{T}}$ of $\mathrm{Gr}_2(\mathbb{C}^n)$ with each trivalent tree $\mathcal{T}$ having $n$ leaves. These degenerations induce toric degenerations $M_{\mathbf{r}}^{\mathcal{T}}$ of $M_{\mathbf{r}}$, the space of $n$ ordered, weighted (by $\mathbf{r}$) points on the projective line. Our goal in this paper is to give a geometric (Euclidean polygon) description of the toric fibers and describe the action of the compact part of the torus as "bendings of polygons". We prove the conjecture of Foth and Hu that the toric fibers are homeomorphic to the spaces defined by Kamiyama and Yoshida.

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.

In this paper, we give several pictorial fractal representations of the ample or Kähler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$.

We give another proof of Vojta's $1+\varepsilon$ conjecture.

Very ampleness criteria for rank $2$ vector bundles over smooth, ruled surfaces over rational and elliptic curves are given. The criteria are then used to settle open existence questions for some special threefolds of low degree.

An inductive approach to classifying all toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are $674,\!688$ such varieties.

Let $\mathcal{F}$ be a coherent rank $2$ sheaf on a scheme $Y \subset \mathbb{P}^{n}$ of dimension at least two and let $X \subset Y$ be the zero set of a section $\sigma \in H^0(\mathcal{F})$. In this paper, we study the relationship between the functor that deforms the pair $(\mathcal{F},\sigma)$ and the two functors that deform $\mathcal{F}$ on $Y$, and $X$ in $Y$, respectively. By imposing some conditions on two forgetful maps between the functors, we prove that the scheme structure of \emph{e.g.,} the moduli scheme ${\rm M_Y}(P)$ of stable sheaves on a threefold $Y$ at $(\mathcal{F})$, and the scheme structure at $(X)$ of the Hilbert scheme of curves on $Y$ become closely related. Using this relationship, we get criteria for the dimension and smoothness of $ {\rm M_{Y}}(P)$ at $(\mathcal{F})$, without assuming $ {\textrm{Ext}^2}(\mathcal{F} ,\mathcal{F} ) = 0$. For reflexive sheaves on $Y=\mathbb{P}^{3}$ whose deficiency module $M = H_{*}^1(\mathcal{F})$ satisfies $ {_{0}\! \textrm{Ext}^2}(M ,M ) = 0 $ (\emph{e.g.,} of diameter at most 2), we get necessary and sufficient conditions of unobstructedness that coincide in the diameter one case. The conditions are further equivalent to the vanishing of certain graded Betti numbers of the free graded minimal resolution of $H_{*}^0(\mathcal{F})$. Moreover, we show that every irreducible component of ${\rm M}_{\mathbb{P}^{3}}(P)$ containing a reflexive sheaf of diameter one is reduced (generically smooth) and we compute its dimension. We also determine a good lower bound for the dimension of any component of ${\rm M}_{\mathbb{P}^{3}}(P)$ that contains a reflexive stable sheaf with ``small'' deficiency module $M$.

We prove that the quantum cohomology ring of any minuscule or cominuscule homogeneous space, specialized at $q=1$, is semisimple. This implies that complex conjugation defines an algebra automorphism of the quantum cohomology ring localized at the quantum parameter. We check that this involution coincides with the strange duality defined in our previous article. We deduce Vafa--Intriligator type formulas for the Gromov--Witten invariants.

A set of necessary and sufficient conditions for the Brascamp--Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank $1$. This complements the result of Barthe concerning the case where the linear maps all have rank $1$. Pushing our analysis further, we describe the case where the maps have either rank $1$ or rank $2$. A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp--Lieb inequality to hold. We present an algorithm which generates such a list.

In this paper we study the supersingular locus of the reduction modulo $p$ of the Shimura variety for $GU(1,s)$ in the case of an inert prime $p$. Using Dieudonné theory we define a stratification of the corresponding moduli space of $p$-divisible groups. We describe the incidence relation of this stratification in terms of the Bruhat--Tits building of a unitary group. In the case of $GU(1,2)$, we show that the supersingular locus is equidimensional of dimension 1 and is of complete intersection. We give an explicit description of the irreducible components and their intersection behaviour.

We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

We use the fixed point arrangement technique developed by Goresky and MacPherson to calculate the part of the equivariant cohomology of the affine flag variety $\mathcal{F}\ell_G$ generated by degree 2. We use this result to show that the vertices of the moment map image of $\mathcal{F}\ell_G$ lie on a paraboloid.

In this paper, we reinterpret the Colmez conjecture on the Faltings height of CM abelian varieties in terms of Hilbert (and Siegel) modular forms. We construct an elliptic modular form involving the Faltings height of a CM abelian surface and arithmetic intersection numbers, and prove that the Colmez conjecture for CM abelian surfaces is equivalent to the cuspidality of this modular form.

If an algebraic torus $T$ acts on a complex projective algebraic variety $X$, then the affine scheme $\operatorname{Spec} H^*_T(X;\mathbb C)$ associated with the equivariant cohomology is often an arrangement of linear subspaces of the vector space $H_2^T(X;\mathbb C).$ In many situations the ordinary cohomology ring of $X$ can be described in terms of this arrangement.

In this work, we investigate how to decompose a pair $(A,B)$ of loxodromic isometries of the complex hyperbolic plane $\mathbf H^{2}_{\mathbb C}$ under the form $A=I_1I_2$ and $B=I_3I_2$, where the $I_k$'s are involutions. The main result is a decomposability criterion, which is expressed in terms of traces of elements of the group $\langle A,B\rangle$.

We give some examples of Calabi--Yau $3$-folds with $\rho=1$ and $\rho=2$, defined over $\mathbb{Q}$ and constructed as $4$-codimensional subvarieties of $\mathbb{P}^7$ via commutative algebra methods. We explain how to deduce their Hodge diamond and top Chern classes from computer based computations over some finite field $\mathbb{F}_{p}$. Three of our examples (of degree $17$ and $20$) are new. The two others (degree $15$ and $18$) are known, and we recover their well-known invariants with our method. These examples are built out of Gulliksen--Neg{\aa}rd and Kustin--Miller complexes of locally free sheaves. Finally, we give two new examples of Calabi--Yau $3$-folds of $\mathbb{P}^6$ of degree $14$ and $15$ (defined over $\mathbb{Q}$). We show that they are not deformation equivalent to Tonoli's examples of the same degree, despite the fact that they have the same invariants $(H^3,c_2\cdot H, c_3)$ and $\rho=1$.

Pour toute sous-vari\'et\'e g\'eom\'etriquement irr\'eductible $V$ du grou\-pe multiplicatif $\mathbb{G}_m^n$, on sait qu'en dehors d'un nombre fini de translat\'es de tores exceptionnels inclus dans $V$, tous les points sont de hauteur minor\'ee par une certaine quantit\'e $q(V)^{-1}>0$. On conna\^it de plus une borne sup\'erieure pour la somme des degr\'es de ces translat\'es de tores pour des valeurs de $q(V)$ polynomiales en le degr\'e de $V$. Ceci n'est pas le cas si l'on exige une minoration quasi-optimale pour la hauteur des points de $V$, essentiellement lin\'eaire en l'inverse du degr\'e. Nous apportons ici une r\'eponse partielle \`a ce probl\`eme\,: nous donnons une majoration de la somme des degr\'es de ces translat\'es de sous-tores de codimension $1$ d'une hypersurface $V$. Les r\'esultats, obtenus dans le cas de $\mathbb{G}_m^3$, mais compl\`etement explicites, peuvent toutefois s'\'etendre \`a $\mathbb{G}_m^n$, moyennant quelques petites complications inh\'erentes \`a la dimension $n$.

We explore the geometric notion of prolongations in the setting of computational algebra, extending results of Landsberg and Manivel which relate prolongations to equations for secant varieties. We also develop methods for computing prolongations that are combinatorial in nature. As an application, we use prolongations to derive a new family of secant equations for the binary symmetric model in phylogenetics.

The theorems of Gross--Zagier and Zhang relate the N\'eron--Tate heights of complex multiplication points on the modular curve $X_0(N)$ (and on Shimura curve analogues) with the central derivatives of automorphic $L$-function. We extend these results to include certain CM points on modular curves of the form $X(\Gamma_0(M)\cap\Gamma_1(S))$ (and on Shimura curve analogues). These results are motivated by applications to Hida theory that can be found in the companion article "Central derivatives of $L$-functions in Hida families", Math.\ Ann.\ \textbf{399}(2007), 803--818.

Let $K$ be a complex reductive algebraic group and $V$ a representation of $K$. Let $S$ denote the ring of polynomials on $V$. Assume that the action of $K$ on $S$ is multiplicity-free. If $\lambda$ denotes the isomorphism class of an irreducible representation of $K$, let $\rho_\lambda\from K \rightarrow GL(V_{\lambda})$ denote the corresponding irreducible representation and $S_\lambda$ the $\lambda$-isotypic component of $S$. Write $S_\lambda \cdot S_\mu$ for the subspace of $S$ spanned by products of $S_\lambda$ and $S_\mu$. If $V_\nu$ occurs as an irreducible constituent of $V_\lambda\otimes V_\mu$, is it true that $S_\nu\subseteq S_\lambda\cdot S_\mu$? In this paper, the authors investigate this question for representations arising in the context of Hermitian symmetric pairs. It is shown that the answer is yes in some cases and, using an earlier result of Ruitenburg, that in the remaining classical cases, the answer is yes provided that a conjecture of Stanley on the multiplication of Jack polynomials is true. It is also shown how the conjecture connects multiplication in the ring $S$ to the usual Littlewood--Richardson rule.

In this paper we prove that a generalized version of the Minimal Resolution Conjecture given by Musta\c{t}\v{a} holds for certain general sets of points on a smooth cubic surface $X \subset \PP^3$. The main tool used is Gorenstein liaison theory and, more precisely, the relationship between the free resolutions of two linked schemes.

Let $X$ be a smooth projective geometrically connected curve over a finite field with function field $K$. Let $\G$ be a connected semisimple group scheme over $X$. Under certain hypotheses we prove the equality of two numbers associated with $\G$. The first is an arithmetic invariant, its Tamagawa number. The second is a geometric invariant, the number of connected components of the moduli stack of $\G$-torsors on $X$. Our results are most useful for studying connected components as much is known about Tamagawa numbers.