Related papers: Cox ring of the generic fiber

We prove that the Cox ring of a smooth rational surface with big anticanonical class is finitely generated. We classify surfaces of this type that are blow-ups of the plane at distinct points lying on a (possibly reducible) cubic.

The aim of this paper is twofold. Firstly, we determine which blow-ups of products of projective spaces at general points are varieties of Fano type, and give boundary divisors making these spaces log Fano pairs. Secondly, we describe…

Let $K$ be a finite extension of the $p$-adic numbers $\mathbb Q_p$ with ring of integers $\mathcal O_K$, $\mathcal X$ a regular scheme, proper, flat, and geometrically irreducible over $\mathcal O_K$ of dimension $d$, and $\mathcal X_K$…

Given a split $\mathbb{P}$-functor $F:\mathcal{D}^b(X) \to \mathcal{D}^b(Y)$ between smooth projective varieties, we provide necessary and sufficient conditions, in terms of the Hochschild cohomology of $X$, for it to become spherical on…

We consider dominant, generically algebraic, and tamely ramified (if the characteristic is positive) morphisms $\pi: X/S \to Y/S$, where Y,S are Noetherian and integral and X is a Krull scheme (e.g. normal Noetherian), and study the sheaf…

We prove conditions under which the total space of the pullback of a sphere fibration over a connected sum is homotopy equivalent to a connected sum with a gyration. Existing results of this type often depend on geometric methods. We…

We provide a simple condition on rational cohomology for the total space of a pullback fibration over a connected sum to have the rational homotopy type of a connected sum, after looping. This takes inspiration from recent work of Jeffrey…

Work of Gonz\'alez, Hering, Payne, and S\"uss shows that it is possible to find both examples and non-examples of Mori dream spaces among projectivized toric vector bundles. This result, and the combinatorial nature of the data of…

Let $X$ be a complex, irreducible, quasi-projective variety, and $\pi:\widetilde X\to X$ a resolution of singularities of $X$. Assume that the singular locus ${\text{Sing}}(X)$ of $X$ is smooth, that the induced map…

We construct a ring structure on complex cobordism tensored with the rationals, which is related to the usual ring structure as quantum cohomology is related to ordinary cohomology. The resulting object defines a generalized two-…

We study generalised Taylor morphisms, functors which construct differential ring homomorphisms from ring homomorphisms in a uniform way, analogous to the Taylor expansion for smooth functions. We generalise the construction of the twisted…

We initiate the study of deformation theory in the context of derived and higher log geometry. After reconceptualizing the "exactification"-procedures in ordinary log geometry in terms of Quillen's approach to the cotangent complex, we…

In this note we show that the nilpotence conjecture for toric varieties is true over any regular coefficient ring containing Q.

We compute the Cox rings of the blow-ups $\mathrm{Bl}_\Delta(X'\times X')$ and $\mathrm{Bl}_\Delta(\mathbb P_1^n)$ where $X'$ is a product of projective spaces and $\Delta$ is the (generalised) diagonal.

We show that the equivariant Chow cohomology ring of a toric variety is naturally isomorphic to the ring of integral piecewise polynomial functions on the associated fan. This gives a large class of singular spaces for which localization…

We examine the cokernel of the canonical homomorphism from the trivial source ring of a finite group to the ring of $p$-rational complex characters. We use Boltje and Co\c{s}kun's theory of fibered biset functors to determine the structure…

We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric…

For a fibration with the fiber $K(\pi,n)$-space, the algebraic model as a twisted tensor product of chains of the base with standard chains of $K(\pi,n)$-complex is given which preserves multiplicative structure as well. In terms of this…

\special{html:<a href="hrefstring">} Let $Y$ be a del Pezzo variety of degree $d\leq 4$ and dimension $n\geq 3$, let $H$ be an ample class such that $-K_Y=(n-1)H$ and let $Z\subset Y$ be a $0$-dimensional subscheme of length $d$ such that…

Let X be a smooth, complete, toric variety. We study those curves C in X that are contractible, in the sense that there exists an equivariant morphism with connected fibers, with source X, that contracts exactly the irreducible curves that…