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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 $mathrm{SL}_2 times mathbb{A}^1$.
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 f
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 $pi : Sigma to X$
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:Xrightarro
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}^1ti
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