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If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow that the morphism is an isomorphism? This problem is called the local isomorphism problem. In this paper, we use jet schemes to introduce various closure operations among ideals and relate them to the local isomorphism problem. This approach leads to a partial solution of the local isomorphism problem, which is shown to have a negative answer in general and a positive one in several situations of geometric interest.
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In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure $overline{B(1)}$ of type A, and study relations between an NCCR and crepant resolutions $Y$ and $Y^+$ of $overline{B(1)}$. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions $Y$ and $Y^+$ of $overline{B(1)}$ as moduli spaces of representations of the quiver. We also study the Kawamata-Namikawas derived equivalence between crepant resolutions $Y$ and $Y^+$ of $overline{B(1)}$ in terms of an NCCR. We also show that the P-twist on the derived category of $Y$ corresponds to a certain operation of the NCCR, which we call multi-mutation, and that a multi-mutation is a composition of Iyama-Wemysss mutations.
We study the Hodge filtration on the local cohomology sheaves of a smooth complex algebraic variety along a closed subscheme Z in terms of log resolutions, and derive applications regarding the local cohomological dimension, the Du Bois complex, local vanishing, and reflexive differentials associated to Z.
Let G be a reductive linear algebraic group, H a reductive subgroup of G and X an affine G-variety. Let Y denote the set of fixed points of H in X, and N(H) the normalizer of H in G. In this paper we study the natural map from the quotient of Y by N(H) to the quotient of X by G induced by the inclusion of Y in X. We show that, given G and H, this map is a finite morphism for all G-varieties X if and only if H is G-completely reducible (in the sense defined by J-P. Serre); this was proved in characteristic zero by Luna in the 1970s. We discuss some applications and give a criterion for the map of quotients to be an isomorphism. We show how to extend some other results in Lunas paper to positive characteristic and also prove the following theorem. Let H and K be reductive subgroups of G; then the double coset HgK is closed for generic g in G if and only if the intersection of generic conjugates of H and K is reductive.
The ACC conjecture for local volumes predicts that the set of local volumes of klt singularities $xin (X,Delta)$ satisfies the ACC if the coefficients of $Delta$ belong to a DCC set. In this paper, we prove the ACC conjecture for local volumes under the assumption that the ambient germ is analytically bounded. We introduce another related conjecture, which predicts the existence of $delta$-plt blow-ups of a klt singularity whose local volume has a positive lower bound. We show that the latter conjecture also holds when the ambient germ is analytically bounded. Moreover, we prove that both conjectures hold in dimension 2 as well as for 3-dimensional terminal singularities.
Let $P_{text{MAX}}(d,s)$ denote the maximum arithmetic genus of a locally Cohen-Macaulay curve of degree $d$ in $mathbb{P}^3$ that is not contained in a surface of degree $<s$. A bound $P(d, s)$ for $P_{text{MAX}}(d,s)$ has been proven by the first author in characteristic zero and then generalized in any characteristic by the third author. In this paper, we construct a large family $mathcal{C}$ of primitive multiple lines and we conjecture that the generic element of $mathcal{C}$ has good cohomological properties. With the aid of emph{Macaulay2} we checked the validity of the conjecture for $s leq 100$. From the conjecture it would follow that $P(d,s)= P_{text{MAX}}(d,s)$ for $d=s$ and for every $d geq 2s-1$.
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