Related papers: Veronese Varieties over Fields with non-zero Chara…

The ideal of a Segre variety is generated by the 2-minors of a generic hypermatrix of indeterminates. We extend this result to the case of Segre-Veronese varieties. The main tool is the concept of weak generic hypermatrix which allows us to…

In this article we characterize the complex hyperbolic groups that leave invariant a copy of the Veronese curve in $\Bbb{P}^2_{\Bbb{C}}$. As a corollary we get that every discrete compact surface group in $\PO^+(2,1)$ admits a deformation…

We study some density results for integral points on the complement of a closed subvariety in the $n$-dimensional projective space over a number field. For instance, we consider a subvariety whose components consist of $n-1$ hyperplanes…

We define new classes of modules of equations for secant varieties of Veronese varieties using representation theory and geometry. We also revisit some old modules of equations (catalecticant minors) to determine when they are sufficient to…

In this paper we describe all possible reduced complete intersection sets of points on Veronese surfaces. We formulate a conjecture for the general case of complete intersection subvarieties of any dimension and we prove it in the case of…

This survey paper discusses some of the recent progress in the study of rational curves on algebraic varieties. It was written for the survey volume of the priority programme "Global Methods in Complex Geometry", supported by the DFG. To…

Let $X$ be a quasi-projective variety over a number field, admitting (after passage to $\mathbb{C}$) a geometric variation of Hodge structure whose period mapping has zero-dimensional fibers. Then the integral points of $X$ are sparse: the…

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in…

Consider a given space, e.g., the Euclidean plane, and its decomposition into Voronoi regions induced by given sites. It seems intuitively clear that each point in the space belongs to at least one of the regions, i.e., no neutral region…

We construct wonderful compactifications of the spaces of linear maps, and symmetric linear maps of a given rank as blow-ups of secant varieties of Segre and Veronese varieties. Furthermore, we investigate their birational geometry and…

We consider the varieties $O_{k,n.d}$ of the k-osculating spaces to the Veronese varieties, the $d-$uple embeddings of $\PP n$; we study the dimension of their higher secant varieties. Via inverse systems (apolarity) and the study of…

Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…

We give an almost asymptotically sharp bound for the non secant defectiveness and identifiability of Segre-Veronese varieties. We also provide new examples of defective Segre-Veronese varieties.

We survey - by means of 20 examples - the concept of varifold, as generalised submanifold, with emphasis on regularity of integral varifolds with mean curvature, while keeping prerequisites to a minimum. Integral varifolds are the natural…

In this paper we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is…

We show that the moduli space of genus zero stable maps is a real projective variety if the target space is a smooth convex real projective variety. We show that evaluation maps, forgetful maps are real morphisms. We analyze the real part…

Let $X(\mathbb {C})\subset \mathbb {P}^r(\mathbb {C})$ be an integral non-degenerate variety defined over $\mathbb {R}$. For any $q\in \mathbb {P}^r(\mathbb {R})$ we study the existence of $S\subset X(\mathbb {C})$ with small cardinality,…

We prove two related concentration inequalities concerning the number of rational points of hyperelliptic curves over subsets of a finite field. In particular, we investigate the probability of a large discrepancy between the numbers of…

Consider a smooth log Fano variety over the function field of a curve. Suppose that the boundary has positive normal bundle. Choose an integral model over the curve. Then integral points are Zariski dense, after removing an explicit finite…

This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical…