Related papers: Factorial threefolds with Ga-actions
The automorphism groups of certain factorial complex affine threefolds admitting locally trivial actions of the additive group are determined. As a consequence new counterexamples to a generalized cancellation problem are obtained.
We construct two non isomorphic contractible affine threefolds X and Y with isomorphic cylinders, showing that the generalized Cancellation Problem has a negative answer in general for contractible affine threefolds. We also establish that…
The cancellation problem asks if two complex algebraic varieties X and Y of the same dimension such that X\times\mathbb{C} and Y\times\mathbb{C} are isomorphic are isomorphic. Iitaka and Fujita established that the answer is positive for a…
We describe a method to construct hypersurfaces of the complex affine $n$-space with isomorphic $\mathbb{C}^*$-cylinders. Among these hypersurfaces, we find new explicit counterexamples to the Laurent Cancellation Problem, i.e.…
Affine varieties of dimension greater than two can be explored their structures with the help of fibrations by the affine line or plane and quotient morphisms by $\mathbb{G}_a$-actions. We consider $\mathbb{G}_a$-actions on affine…
We give a corrected statement of the theorem of Gurjar and Miyanishi, which classifies smooth affine surfaces of Kodaira dimension zero, whose coordinate ring is factorial and has trivial units. Denote the class of such surfaces by…
We address a variant of Zariski Cancellation Problem, asking whether two varieties which become isomorphic after taking their product with an algebraic torus are isomorphic themselves. Such cancellation property is easily checked for…
Let $X$ and $X'$ be affine algebraic varieties over a field $\mathbb{k}$. The celebrated Zariski Cancellation Problem asks as to when the existence of an isomorphism $X\times\mathbb{A}^n\cong X'\times\mathbb{A}^n$ implies $X\cong X'$. In…
This paper considers the family $\mathscr{S}_0$ of smooth affine factorial surfaces of logarithmic Kodaira dimension 0 with trivial units over an algebraically closed field $k$. Our main result (Theorem 4.1) is that the number of…
We give a criterion of factoriality of a suspension. This allows to construct many examples of flexible affine factorial varieties. In particular, we find a homogeneous affine factorial 3-fold that is not a homogeneous space of an algebraic…
A classical result of Miyanishi-Sugie and Keel-McKernan asserts that for smooth affine surfaces, affine-uniruledness is equivalent to affine-ruledness, both properties being in fact equivalent to the negativity of the logarithmic Kodaira…
We consider algebraic manifolds $Y$ of dimension 3 over $\Bbb{C}$ with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$ and $i>0$. Let $X$ be a smooth completion of $Y$ with $D=X-Y$, an effective divisor on $X$ with normal crossings. If the…
We give families of examples of principal open subsets of the affine space \mathbb{A}^{3} which do not have the cancellation property. We show as a by-product that the cylinders over Koras-Russell threefolds of the first kind have a trivial…
Motivated by the study of the structure of algebraic actions the additive group on affine threefolds X, we consider a special class of such varieties whose algebraic quotient morphisms X $\rightarrow$ X//Ga restrict to principal homogeneous…
We study the behavior of the Kodaira dimension of algebraic fiber spaces over threefolds. We prove some cases of the Iitaka Conjecture $C_{n,3}$, including certain situations where the base variety is a Calabi--Yau threefold.
Every $\mathbb{A}^{1}-$bundle over the complex affine plane punctured at the origin, is trivial in the differentiable category but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of…
We analyze the structure of the algebraic manifolds $Y$ of dimension 3 with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$, by showing the deformation invariant of some open surfaces. Secondly, we show…
We construct algebraic families of smooth affine $\mathbb{A}^1$-contractible varieties of every dimension $n\geq 4$ over fields of characteristic zero which are non-isomorphic to affine spaces and potential counterexamples to the Zariski…
We study a two-dimensional family of affine surfaces which are counter-examples to the Cancellation Problem. We describe the Makar-Limanov invariant of these surfaces, determine their isomorphism classes and characterize the automorphisms…
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.