Related papers: Partially complex ranks for real projective variet…

Let $(G, \omega)$ be a hyperelliptic vertex-weighted graph of genus $g \geq 2$. We give a characterization of $(G, \omega)$ for which there exists a smooth projective curve $X$ of genus $g$ over a complete discrete valuation field with…

We present a new, far simpler family of counter-examples to Kushnirenko's Conjecture. Along the way, we illustrate a computer-assisted approach to finding sparse polynomial systems with maximally many real roots, thus shedding light on the…

Let $K$ be a non-empty set of ideals of the commutative ring $R$, closed under taking smaller ideals. A subset $X$ of the group ring $R[\mathbb{Z}^s]$ is called a $K$-set if the ideal generated by the coefficients of the elements of $X$ is…

We construct a projective variety with discrete, non-finitely generated automorphism group. As an application, we show that there exists a complex projective variety with infinitely many non-isomorphic real forms.

Let V be a smooth equidimensional quasi-affine variety of dimension r over the complex numbers $C$ and let $F$ be a $(p\times s)$-matrix of coordinate functions of $C[V]$, where $s\ge p+r$. The pair $(V,F)$ determines a vector bundle $E$ of…

We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics we…

In genetic studies, not only can the number of predictors obtained from microarray measurements be extremely large, there can also be multiple response variables. Motivated by such a situation, we consider semiparametric dimension reduction…

A semiholomorphic foliations of type (n, d) is a differentiable real manifold X of dimension 2n + d, foliated by complex leaves of complex dimension n. In the present work, we introduce an appropriate notion of pseudoconvexity (and…

We study families of linear spaces in projective space whose union is a proper subvariety X of the expected dimension. We establish relations between configurations of focal points and existence or non-existence of a fixed tangent space to…

For an interval finite quiver $Q$, we introduce a class of flat representations. We classify the indecomposable projective objects in the category $\mathrm{rep}(Q)$ of pointwise finite dimensional representations. We show that an object in…

We study the arithmetic properties of projective varieties of almost minimal degree, that is of non-degenerate irreducible projective varieties whose degree exceeds the codimension by precisely 2. We notably show, that such a variety $X…

For certain problems involving vector fields, it is possible to find an associated imaginary field that, in conjunction with the first, forms a complex field for which the equation can be solved. This result is generalized to arbitrary…

An n-dimensional submanifold X of a projective space P^N (C) is called tangentially degenerate if the rank of its Gauss mapping \gamma: X ---> G (n, N) satisfies 0 < rank \gamma < n. The authors systematically study the geometry of…

Let $(X,L,V)$ be a triplet where $X$ is an irreducible smooth complex projective variety, $L$ is an ample and spanned line bundle on $X$ and $V\subseteq H^0(X,L)$ spans $L$. The discriminant locus $\Cal D(X,V) \subset |V|$ is the algebraic…

Various aspects of orbifolds and cosets of the small $\mathcal{N}=4$ superconformal algebra are studied. First, we determine minimal strong generators for generic and specific levels. As a corollary, we obtain the vertex algebra of global…

We characterize $q$-ample Ulrich bundles on a variety $X \subseteq \mathbb P^N$ with respect to $(q+1)$-dimensional linear spaces contained in $X$.

We prove that the abundance conjecture holds on a variety $X$ with mild singularities if $X$ has many reflexive differential forms with coefficients in pluricanonical bundles, assuming the Minimal Model Program in lower dimensions. This…

Let $X\subseteq \mathbb{P}^m$ be a totally real, non-degenerate, projective variety and let $\Gamma\subseteq X(\mathbb{R})$ be a generic set of points. Let $P$ be the cone of nonnegative quadratic forms on $X$ and let $\Sigma$ be the cone…

We show that groups with a mild form of non-positive curvature (a navigable path system) satisfy the weak rank rigidity conjecture: they either have linear divergence or a Morse element. This class includes discrete groups of projective…

Let $V$ be a left vector space over a division ring and let ${\mathcal P}(V)$ be the associated projective space. We describe all finite subsets $X\subset V$ such that every permutation on $X$ can be extended to a linear automorphism of $V$…