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Register a new userDeformations of $mathbb{A}^1$-cylindrical varieties
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An algebraic variety is called $mathbb{A}^{1}$-cylindrical if it contains an $mathbb{A}^{1}$-cylinder, i.e. a Zariski open subset of the form $Ztimesmathbb{A}^{1}$ for some algebraic variety Z. We show that the generic fiber of a family $f:Xrightarrow S$ of normal $mathbb{A}^{1}$-cylindrical varieties becomes $mathbb{A}^{1}$-cylindrical after a finite extension of the base. Our second result is a criterion for existence of an $mathbb{A}^{1}$-cylinder in X which we derive from a careful inspection of a relative Minimal Model Program ran from a suitable smooth relative projective model of X over S.
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We give a general structure theorem for affine A 1-fibrations on smooth quasi-projective surfaces. As an application, we show that every smooth A 1-fibered affine surface non-isomorphic to the total space of a line bundle over a smooth affine curve fails the Zariski Cancellation Problem. The present note is an expanded version of a talk given at the Kinosaki Algebraic Geometry Symposium in October 2019.
We show that the infinitesimal deformations of the Brill--Noether locus $W_d$ attached to a smooth non-hyperelliptic curve $C$ are in one-to-one correspondence with the deformations of $C$. As an application, we prove that if a Jacobian $J$ deforms together with a minimal cohomology class out the Jacobian locus, then $J$ is hyperelliptic. In particular, this provides an evidence to a conjecture of Debarre on the classification of ppavs carrying a minimal cohomology class. Finally, we also study simultaneous deformations of Fano surfaces of lines and intermediate Jacobians.
The main results of this paper are already known (V.V. Shokurov, the non-vanishing theorem, 1985). Moreover, the non-$mathbb{Q}$-factorial MMP was more recently considered by O~Fujino, in the case of toric varieties (Equivariant completions of toric contraction morphisms, 2006), for klt pairs (Special termination and reduction to pl flips, 2007) and more generally for log-canonical pairs (Foundation of the minimal model program, 2014). Here we rewrite the proofs of some of these results, by following the proofs given by Y. Kawamata, K. Matsuda, and K. Matsuki (Introduction to the minimal model problem, 1985) of the same results in $mathbb{Q}$-factorial MMP. And, in the family of $mathbb{Q}$-Gorenstein spherical varieties, we answer positively to the questions of existence of flips and of finiteness of sequences of flips. I apologize for the first version of this paper, which I wrote without knowing that these results already exist.
We reproduce the quantum cohomology of toric varieties (and of some hypersurfaces in projective spaces) as the cohomology of certain vertex algebras with differential. The deformation technique allows us to compute the cohomology of the chiral de Rham complex over the projective space.
In this paper we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimensional two arithmetically Cohen-Macaulay (ACM) varieties in $mathbb P^1timesmathbb P^1timesmathbb P^1$, called varieties of lines. We also describe their ACM property from combinatorial algebra point of view.
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