arXiv subjects

Mircea Mustata

Publications and source records attributed to Mircea Mustata.

78 records · Page 5Linked to original sources

arXiv2001

The Multiplier Ideals of a Sum of Ideals

Let X be a smooth variety and J, K two ideal sheaves on X. We prove the following formula relating the multiplier ideals of J, K and J+K: I(X, c(J+K))\subset \sum_{a+b=c} I(X, aJ)\cdot I(X,bK). An analogous formula holds for the asymptotic multiplier ideals of two graded systems of ideals. As a consequence, we show how to approximate at any given point arbitrary multiplier ideals by multiplier ideals of zero dimensional ideals. We apply this to study the invariance of multiplier ideals under different embeddings.

math.AG↗

arXiv2001

Singularities of Pairs via Jet Schemes

Let X be a smooth variety and Y a closed subscheme of X. By comparing motivic integrals on X and on a log resolution of (X,Y), we prove the following formula for the log canonical threshold of (X,Y): c(X,Y)=dim X-sup_m{(dim Y_m}/(m+1)}, where Y_m is the mth jet scheme of Y. We show how this formula can be used to study the log canonical threshold. In particular, we give a proof of the Semicontinuity theorem of Demailly and Kollár.

math.AG↗

arXiv2000

D-modules on Smooth Toric Varieties

Let X be a smooth toric variety. David Cox introduced the homogeneous coordinate ring S of X and its irrelevant ideal B. Extending well-known results on projective space, Cox established the following: (1) the category of quasi-coherent sheaves on X is equivalent to the category of graded S-modules modulo B-torsion, (2) the variety X is a geometric quotient of Spec(S) V(B) by a suitable torus action. We provide the D-module version of these results. More specifically, let A denote the ring of differential operators on Spec(S). We show that the category of D-modules on X is equivalent to a subcategory of graded A-modules modulo B-torsion. Additionally, we prove that the characteristic variety of a D-module is a geometric quotient of an open subset of the characteristic variety of the associated A-module and that holonomic D-modules correspond to holonomic A-modules.

math.AG↗

arXiv2000

The Module of Logarithmic p-forms of a Locally Free Arrangement

For an essential, central hyperplane arrangement A in V=k^{n+1}, we show that Ω^1(A) (the module of logarithmic one forms with poles along A) gives rise to a locally free sheaf on P^n if and only if for all X in L_A with rank X<dim V, the module Ω^1(A_X) is free. Our main result is that in this case the Poicare polynomial of A is essentially the Chern polynomial. The proof is based on a result of Solomon and Terao and on a formula we give for the Chern polynomial of a bundle E on P^n in terms of the Hilbert series of \oplus_m H^0(\wedge^iE(m)). If Ω^1(A)has projective dimension one and is locally free, we give a minimal free resolution for Ω^p, and show that \wedge^p(Ω^1(A))\isoΩ^p(A), generalizing results of Rose and Terao on generic arrangements.

math.AG↗

arXiv2000

Cohomology on Toric Varieties and Local Cohomology with Monomial Supports

We study the local cohomology modules H^i_B(R) for a reduced monomial ideal B in a polynomial ring R=k[X_1,...,X_n]. We consider a grading on R which is coarser than the Z^n-grading such that each component of H^i_B(R) is finite dimensional and we give an effective way to compute these components. Using Cox's description for sheaves on toric varieties, we apply these results to compute the cohomology of coherent sheaves on toric varieties. We give algorithms for this computation which have been implemented in the Macaulay 2 system. We obtain also a topological description for the cohomology of rank one torsionfree sheaves on toric varieties.

math.AG↗

arXiv2000

Local Cohomology at Monomial Ideals

For a reduced monomial ideal B in R=k[X_1,...,X_n], we write H^i_B(R) as the union of {Ext^i(R/B^[d],R)}_d, where {B^[d]}_d are the "Frobenius powers of B". We describe H^i_B(R)_p, for every p in Z^n, in the spirit of the Stanley-Reisner theory. As a first application we give an isomorphism Tor_i(B', k)_p\iso Ext^{|p|-i}(R/B,R)_{-p} for all p in {0,1}^n, where B' is the Alexander dual ideal of B. We deduce a canonical filtration of Ext^i(R/B,R) with succesive quotients of the form R/(X_{j_1},...,X_{j_i}) suitably shifted, the multiplicities being computed from the Betti numbers of B'. As a final application, we give a topological description for the associated primes of Ext^i(R/B,R).

math.AC↗