Related papers: A Phantom on a Rational Surface

Let $L$ be a simply-connected simple connected algebraic group over a number field $F$, and $H$ be a semisimple absolutely maximal connected $F$-subgroup of $L$. Under a cohomological condition, we prove an asymptotic formula for the number…

In the setting of symplectic manifolds which are convex at infinity, we use a version of the Aleksandrov maximum principle to derive uniform estimates for Floer solutions that are valid for a wider class of Hamiltonians and almost complex…

We investigate when the tangent bundle of a projective manifold has a non-trivial first order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.

A partial affine plane of order $n$ is a point-line incidence structure with $n^2$ points and $n$ points on each line, such that every two lines meet in at most one point. In this paper, we show that a partial affine plane of order $n$, $n$…

This article provides a new perspective on the geometry of a projective line, which helps clarify and illuminate some classical results about projective plane. As part of the same train of ideas, the article also provides a proof of the…

Many examples of nonpositively curved closed manifolds arise as blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W, and the blow-up locus is W-invariant, then the…

In this note, we continue the study of Seshadri constants on blow-ups of Hirzebruch surfaces initiated in arXiv:2312.14555. Now we consider blow-ups of ruled surfaces more generally. We propose a conjecture for classifying all the negative…

Rapha\"{e}l Rouquier introduced an invariant of triangulated categories which is known as Rouquier dimension. Orlov conjectured that for any smooth quasi-projective variety $X$ the Rouquier dimension of $D^b_{\mathrm{coh}}(X)$ is equal to…

We propose a general definition of mathematical instanton bundle with given charge on any Fano threefold extending the classical definitions on $\mathbb P^3$ and on Fano threefold with cyclic Picard group. Then we deal with the case of the…

We study k-jet ampleness of line bundles on hyperelliptic surfaces using vanishing theorems. Our main result states that on a hyperelliptic surface of an arbitrary type a line bundle of type (m,m) with m\geq k+2 is k-jet ample.

We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.

We find minimal and maximal length of intersections of lines at a fixed distance to the origin with the cross-polytope. We also find maximal volume noncentral sections of the cross-polypote by hyperplanes which are at a fixed large distance…

We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal non-percolating sets. Additional…

We address the problem of bounding from below the self-intersection of integral curves on the projective plane blown-up at general points. In particular, by applying classical deformation theory we obtain the expected bound in the case of…

We discuss homogeneity and universality issues in the theory of abstract linear spaces, namely, structures with points and lines satisfying natural axioms, as in Euclidean or projective geometry. We show that the two smallest projective…

We investigate cosmological scenarios with a non-minimal derivative coupling between the scalar field and the curvature, examining both the quintessence and the phantom cases in zero and constant potentials. In general, we find that the…

The projective variety of square-zero elements in the six-dimensional minimal supersymmetry algebra is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^3$. We use this fact, together with the pure spinor superfield formalism, to study…

We consider the set of points in projective $n$-space that generate an extension of degree $e$ over given number field $k$, and deduce an asymptotic formula for the number of such points of absolute height at most $X$, as $X$ tends to…

It contains the proof of a very general $\partial\bar\partial$-lemma, together with a decomposition theorem for currents with values in a (singular) Hermitian line bundle. As a corollary, we establish the K\"ahler version on an injectivity…

Erd\H{o}s and Fishburn studied the maximum number of points in the plane that span $k$ distances and classified these configurations, as an inverse problem of the Erd\H{o}s distinct distances problem. We consider the analogous problem for…