Related papers: Sharp volume and multiplicity bounds for Fano simp…

We get sharp degree bound for generic smoothness and connectedness of the space of conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on Hypersurfaces.

For $n\geq 4$, let $X$ be a complex smooth Fano $n$-fold whose minimal anticanonical degree of non-free rational curves on $X$ is at least $n-2$. We classify extremal contractions of such varieties. As an application, we obtain a…

We provide enumerative formulas for the degrees of varieties parameterizing hypersurfaces and complete intersections which contain pro-jective subspaces and conics. Besides, we find all cases where the Fano scheme of the general complete…

We construct an explicit lower bound for the volume of a complex hyperbolic orbifold that depends only on dimension.

We study global log canonical thresholds on anticanonically embedded quasismooth weighted Fano threefold hypersurfaces having terminal quotient singularities to prove the existence of a Kahler-Einstein metric on most of them, and to produce…

Determining the number of embeddings of Laman graph frameworks is an open problem which corresponds to understanding the solutions of the resulting systems of equations. In this paper we investigate the bounds which can be obtained from the…

The classification of toric Fano manifolds with large Picard number corresponds to the classification of smooth Fano polytopes with large number of vertices. A smooth Fano polytope is a polytope that contains the origin in its interior such…

We introduce a manifestly little group covariant on-shell superspace for massive particles in four dimensions using the massive spinor helicity formalism. This enables us to construct massive on-shell superfields and fully utilize on-shell…

We prove tight upper bounds for the number of vertices of a simple polygon that is the union or the intersection of two simple polygons with given numbers of convex and concave vertices. The similar question on graphs of the lower (or…

We introduce complex singularity exponents of plurisubharmonic functions and prove a general semi-continuity result for them. This concept contains as a special case several similar concepts which have been considered e.g. by Arnold and…

We study Fano manifolds of pseudoindex greater than one and dimension greater than five, which are blow-ups of smooth varieties along smooth centers of dimension equal to the pseudoindex of the manifold. We obtain a classification of the…

In this article we provide lower bounds for the lower Hausdorff dimension of finite measures assuming certain restrictions on their quaternionic spherical harmonics expansion. This estimate is an analog of a result previously obtained by…

Let $\mathcal{T}^d(1)$ be the set of all $d$-dimensional simplices $T$ in $\real^d$ with integer vertices and a single integer point in the interior of $T$. It follows from a result of Hensley that $\mathcal{T}^d(1)$ is finite up to affine…

We obtain upper bounds for the multiplicity of an isolated solution of a system of equations $f_1=...= f_M =0$ in $M$ variables, where the set of polynomials $(f_1,..., f_M)$ is a tuple of general position in a subvariety of a given…

We investigate Fano varieties defined over a number field that contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety.

We provide a complete classification of Fano threefolds X having canonical Gorenstein singularities and the anticanonical degree (-KX)^3 equal 64.

We classify Fano polygons with finite mutation class. This classification exploits a correspondence between Fano polygons and cluster algebras, refining the notion of singularity content due to Akhtar and Kasprzyk. We also introduce…

We describe the moduli space of rational curves on smooth Fano varieties of coindex 3. For varieties of dimension 5 or greater, we prove the moduli space has a single irreducible component for each effective numerical class of curves. For…

For any positive integer $k$ and any integer $n$ large enough, we construct a Fano variety $X$ with Picard number $k$ and dimension $n$ such that $((-K_X)^n)^{1/n}$ grows like $n^k/(\log n)^{k-1}$.

In this article we prove an upper bound for a Hilbert polynomial on quaternionic Kaehler manifolds of positive scalar curvature. As corollaries we obtain bounds on the quaternionic volume and the degree of the associated twistor space.…