Related papers: Rouquier dimension of some blow-ups

Utilizing a splitting of geometric flows on surfaces introduced by Buzano and Rupflin, we present a general scheme to prove blow up criteria for such geometric flows. A vital ingredient is a new compactness theorem for families of metrics…

The following conjecture arose out of discussions between B. Harbourne, J. Ro\'e, C. Cilberto and R. Miranda: for a smooth projective surface $X$ there exists a positive constant $c_X$ such that $h^1(\mathcal O_X(C))\le c_X h^0(\mathcal…

It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of…

We consider diagonal cubic surfaces defined by an equation of the form ax^3+by^3+cz^3+dt^3 = 0. Numerically, one can find all rational points of height < B for B in the range of up to 100 000, thanks to a program due to D. J. Bernstein. On…

Suppose $R$ is an excellent ring of dimension $3$ and has rational singularities. Let $\pi:X \longrightarrow \mathrm{Spec} \ R$ be a blow-up and $\phi: W \longrightarrow X$ be any projective, birational morphism such that $X$ and $W$ are…

We prove that the bounded derived category of coherent sheaves with proper support is equivalent to the category of locally-finite, cohomological functors on the perfect derived category of a quasi-projective scheme over a field. We…

We consider equivariant wave maps from $\mathbb{R}^{d+1}$ to $\mathbb{S}^d$ in supercritical dimensions $3\leq d\leq 6$. Using mixed numerical and analytic methods, we show that the threshold of blowup is given by the codimension-one stable…

In this article, we prove that the blow-up of a locally irreducible lcK space $X$ along a subspace $Z$ which verifies certain conditions is lcK if and only if $X$ is induced gcK, generalizing a theorem of Ornea-Verbitsky-Vuletescu to…

We establish a semi-orthogonal decomposition for the weighted blowup of an algebraic stack along a Koszul-regular weighted centre, generalising the classic result of Orlov. Our approach is based on the work of Bergh-Schn\"urer.

Let $G$ be a connected algebraic group. We study $G$-equivariant extremal contractions whose centre is a codimension three $G$-simply connected orbit. In the spirit of an important result by Kawakita in 2001, we prove that those…

Let $X$ be a compact K\"ahler fourfold with klt singularities and vanishing first Chern class, smooth in codimension two. We show that $X$ admits a Beauville-Bogomolov decomposition: a finite quasi-\'etale cover of $X$ splits as a product…

In this article we prove several new uniform upper bounds on the number of points of bounded height on varieties over $\mathbb{F}_q[t]$. For projective curves, we prove the analogue of Walsh' result with polynomial dependence on $q$ and the…

It is a classical result, due to F. Tricceri, that the blow-up of a manifold of locally conformally K\"ahler (l.c.K. for short) type at some point is again of l.c.K. type. However, the proof given in \cite{Tric} is somehow unclear. We give…

We provide a characterization of asymptotical speciality of a nef and big divisor $D$ on an algebraic surface in terms of the arithmetic genus of curves in $D^{\perp}$. As a consequence we prove that the SHGH conjecture for linear systems…

Let U be the tautological subbundle on the Grassmannian $\mathrm{Gr}(k, n)$. There is a natural morphism $\mathrm{Tot}(U) \to \mathbb{A}^n$. Using it, we give a semiorthogonal decomposition for the bounded derived category…

For the critical focusing wave equation \Box u = u^5 on R^{3+1} in the radial case, we prove the existence of type II blow up solutions with scaling parameter \lambda(t) = t^{-1-\nu} for all \nu >0. This extends the previous work by the…

We define the Weyl cycles on $X^n_s$, the blown up projective space $\mathbb{P}^n$ in $s$ points in general position. In particular, we focus on the Mori Dream spaces $X^3_7$ and $X^{4}_{8}$, where we classify all the Weyl cycles of…

We define a dimension for a triangulated category. We prove a representabilityTheorem for a certain class of functors on finite dimensional triangulatedcategories. We study the dimension of the boundedderived category of an algebra or a…

Two conjectures relating the Kodaira dimension of a smooth projective variety and existence of number of nowhere vanishing 1-forms on the variety are proposed and verified in dimension 3.

A random set of points in Euclidean space is called `rigid' or `hyperuniform' if the number of points falling inside any given region has significantly smaller fluctuations than the corresponding number for a set of i.i.d. random points.…