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Register a new userExotic Ga-quotients of SL$_2 times mathbb{A}^1$
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Authors
Adrien Dubouloz
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Every deformed Koras-Russell threefold of the first kind $Y = left{ x^{n}z=y^{m}-t^{r} + xh(x,y,t)right}$ in $mathbb{A}^{4}$ is the algebraic quotient of proper Zariski locally trivial $mathbb{G}_a$-action on $mathrm{SL}_2 times mathbb{A}^1$.
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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 prove a functorial correspondence between a category of logarithmic $mathfrak{sl}_2$-connections on a curve $X$ with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover $pi : Sigma to X$. The proof is by constructing a pair of inverse functors $pi^{text{ab}}, pi_{text{ab}}$, and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor $pi_ast$.
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.
We provide a number of new conjectures and questions concerning the syzygies of $mathbb{P}^1times mathbb{P}^1$. The conjectures are based on computing the graded Betti tables and related data for large number of different embeddings of $mathbb{P}^1times mathbb{P}^1$. These computations utilize linear algebra over finite fields and high-performance computing.
In this paper we prove the equivalence among (i) the weakly coupled worldsheet string theory described by the coset sigma model $frac{SL(2,mathbb{R})_ktimes U(1)}{U(1)}times S^3 times T^4$ with $SL(2,mathbb{R})$ WZW level $kgeq 2$, (ii) the full near horizon theory of the NS5 branes with $k$ NS5 branes wrapping $T^4times S^1$, $pgg1$ F1 strings wrapping $S^1$ and $n$ units of momentum along the $S^1$ and (iii) the single trace $Tbar{T}$ deformation of string theory in $AdS_3times S^3times T^4$. As a check we compute the spectrum of the spacetime theory by performing BRST quantization of the coset description of the worldsheet theory and show that it matches exactly with the one derived in the case of single trace $Tbar{T}$ deformed string theory in $AdS_3$. Secondly, we compute the two-point correlation function of local operators of the spacetime theory using the worldsheet coset approach and reproduce the same two-point function from the supergravity approach.
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