Related papers: The diminished base locus is not always closed

We complete the explicit study of a three-fold divisorial contraction whose exceptional divisor contracts to a point, by treating the case where the point downstairs is a singularity of index $n \ge 2$. We prove that if this singularity is…

We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse…

This is an expository paper. It is not difficult to see that every group homomorphism from the additive group Z of integers to the additive group R of real numbers extends to a homomorphism from R to R. We discuss other examples of discrete…

We show that the Zariski closure of the set of hypersurfaces of degree $M$ in ${\mathbb P}^{M}$, where $M\geq 5$, which are either not factorial or not birationally superrigid, is of codimension at least $\binom{M-3}{2}+1$ in the parameter…

We study the deterministic recursion $n_{j+1} = n_j - \tau(n_j)$, where $\tau(n)$ denotes the divisor function, and the associated orbit length $a(x)$. Heuristics based on the average order of $\tau(n)$ suggest that $a(x) \asymp x / \log…

The decomposition theorem for smooth projective morphisms $\pi:\mathcal{X}\rightarrow B$ says that $R\pi_*\mathbb{Q}$ decomposes as $\oplus R^i\pi_*\mathbb{Q}[-i]$. We describe simple examples where it is not possible to have such a…

Let $S$ be an elliptic surface over a smooth curve $C$ with a section $O$. We denote its generic fiber by $E_S$. For a divisor $D$ on $S$, we canonically associate a $C(C)$-rational point $P_D$. In this note, we give a description of $P_D$…

We study M-separability as well as some other combinatorial versions of separability. In particular, we show that the set-theoretic hypothesis b=d implies that the class of selectively separable spaces is not closed under finite products,…

The space of degree d single-variable monic and centered complex polynomial vector fields can be decomposed into loci in which the vector fields have the same topological structure. We analyze the geometric structure of these loci and…

Let $S$ be a finite set of primes. For sufficiently large $n$ and $d$, Lawrence and Venkatesh proved that in the moduli space of hypersurfaces of degree $d$ in $\mathbb{P}^n$, the locus of points with good reduction outside $S$ is not…

For each integer $d\geq 2$, let $M_d$ denote the moduli space of maps $f: \mathbb{P}^1\to \mathbb{P}^1$ of degree $d$. We study the geometric configurations of subsets of postcritically finite (or PCF) maps in $M_d$. A complex-algebraic…

In this article, we establish a connection between two models for $r$-spin structures on surfaces: the marked PLCW decompositions of Novak and Runkel-Szegedy, and the structured graphs of Dyckerhoff-Kapranov. We use these models to describe…

We study cones of pseudoeffective cycles on the blow up of $({\mathbb P}^1)^n$ at points in very general position, proving some results concerning their structure. In particular we show that in some cases they turn out to be generated by…

Let $G$ be a linear algebraic group over a field. We show that, under mild assumptions, in a family of primitive generically free $G$-varieties over a base variety $B$ the essential dimension of the geometric fibers may drop on a countable…

We explore very stable and wobbly bundles, twisted in a particular sense by a line bundle, over complex algebraic curves of genus $1$. We verify that twisted stable bundles on an elliptic curve are not very stable for any positive twist. We…

The demiclosedness principle is one of the key tools in nonlinear analysis and fixed point theory. In this note, this principle is extended and made more flexible by two mutually orthogonal affine subspaces. Versions for finitely many…

By a theorem of Reider, a twisted bicanonical system, that means a linear system of divisors numerically equivalent to a bicanonical divisor, on a minimal surface of general type, is base point free if $K^2_S \geq 5$. Twisted bicanonical…

In this paper we will think of certain abelian categories with favorable properties as non-commutative surfaces. We show that under certain conditions a point on a non-commutative surface can be blown up. This yields a new non-commutative…

We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless its Zariski closure acts transitively on a Hermitian symmetric space of tube type. This is a kind of converse to a…

We construct a continuum of non-homeomorphic compact subspaces of the real line R without singleton components. Thus from the purely topological point of view the real line contains not only more closed sets than open sets but also more…