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We have proved the following Problem: Let $R$ be a $mathbb{C}$-affine domain, let $T$ be an element in $R setminus mathbb{C}$ and let $i : mathbb{C}[T] hookrightarrow R$ be the inclusion. Assume that $R/TR cong_{mathbb{C}} mathbb{C}^{[n-1]}$ and that $R_T cong_{mathbb{C}[T]} mathbb{C}[T]_T^{[n-1]}$. Then $R cong_{mathbb{C}} mathbb{C}^{[n]}$.} This result leads to the negative solution of the candidate counter-example of V.Arno den Essen : Conjecture E : Let $A:=mathbb{C}[t,u,x,y,z]$ denote a polynomial ring, and let $f(u), g(u)$ and $h(u)$ be the polynomials defined above. Let $D:= f(u)partial_x + g(u)partial_y + h(u)partial_z + tpartial_u$ (which is easily seen to be a locally nilpotent derivation on $A$). Then $A^D otcong_{mathbb{C}} mathbb{C}^{[4]}$.} Consequently it gurantees that the conjectures : the Cancellation Problem for affine spaces, the Linearization Problem and the Embedding Problem and the affine $mathbb{A}^n$-Fibration Problem are still open.
For complete intersection Calabi-Yau manifolds in toric varieties, Gross and Haase-Zharkov have given a conjectural combinatorial description of the special Lagrangian torus fibrations whose existence was predicted by Strominger, Yau and Zaslow. We p
Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R, )$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $ $-stable filtrations ${mathbb M} $ of $M $
Let $(A, m, k)$ be a Gorenstein local ring of dimension $ dgeq 1.$ Let $I$ be an ideal of $A$ with $htt(I) geq d-1.$ We prove that the numerical function [ n mapsto ell(ext_A^i(k, A/I^{n+1}))] is given by a polynomial of degree $d-1 $ in the case w
Filip showed that there are constants $C>0$ and $delta>0$ such that the number of special Lagrangian fibrations of volume $leq V$ in a generic twistor family of K3 surfaces is $Ccdot V^{20}+O(V^{20-delta})$. In this note, we show that $delta$ can b
We give a complete description of the cone of Betti diagrams over a standard graded hypersurface ring of the form k[x,y]/<q>, where q is a homogeneous quadric. We also provide a finite algorithm for decomposing Betti diagrams, including diagrams of i