This is a preliminary draft of monograph. It builds on lectures notes on a course that Lawrence E... more This is a preliminary draft of monograph. It builds on lectures notes on a course that Lawrence Ein gave at the University of Catania in Summer 1998, and later again at Hong Kong University in Fall 1999, on lecture notes from the courses that Tommaso de Fernex taught at the University of Utah in Fall 2006, Spring 2010, and Spring 2012, and on lecture notes for the courses taught by Mircea Mustaţȃ in Winter and Fall 2013 at University of Michigan. This draft has been typeset using an edited version of the Springer Monograph class svmono.cls.
We consider pairs (X, A), where X is a variety with klt singularities and A is a formal product o... more We consider pairs (X, A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also assume that X has (formally) bounded singularities, in the sense that it is, formally locally, a subvariety in a fixed affine space defined by equations of bounded degree. We prove in this context a conjecture of Shokurov, predicting that the set of log canonical thresholds for such pairs satisfies the Ascending Chain Condition.
Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimensi... more Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimension satisfies the ascending chain condition. In this paper we prove that the conjecture holds for log canonical thresholds on smooth varieties and, more generally, on locally complete intersection varieties and on varieties with quotient singularities.
Using inversion of adjunction, we deduce from Nadel's theorem a vanishing property for ideals she... more Using inversion of adjunction, we deduce from Nadel's theorem a vanishing property for ideals sheaves on projective varieties, a special case of which recovers a result due to Bertram-Ein-Lazarsfeld. This enables us to generalize to a large class of projective schemes certain bounds on Castelnuovo-Mumford regularity previously obtained by Bertram-Ein-Lazarsfeld in the smooth case and by Chardin-Ulrich for locally complete intersection varieties with rational singularities. Our results are tested on several examples.
As is well known, the "usual discrepancy" is defined for a normal Q-Gorenstein variety. By using ... more As is well known, the "usual discrepancy" is defined for a normal Q-Gorenstein variety. By using this discrepancy we can define a canonical singularity and a log canonical singularity. In the same way, by using a new notion, Mather-Jacobian discrepancy introduced in recent papers we can define a "canonical singularity" and a "log canonical singularity" for not necessarily normal or Q-Gorenstein varieties. In this paper, we show basic properties of these singularities, behavior of these singularities under deformations and determine all these singularities of dimension up to 2.
The problem of bounding the “complexity" of a polynomial ideal in terms of the degrees of it... more The problem of bounding the “complexity" of a polynomial ideal in terms of the degrees of its generators has attracted a great deal of interest in recent years. Results in this direction go back at least as far as the classical work [17] of Hermann on the ideal membership problem, and the effective Nullstellensatz of Brownawell [4] and Koll ár [21] marks amajor recent advance.With the development of computational algebraic geometry the question has taken on increasing importance, and it came into particularly clear focus through the influential paper [3] of Bayer andMumford.More recently, the theoremof [8] and [20] concerning regularity of powers raises the question of bounding the complexity of powers of an ideal, and suggests that asymptotically the picture should become very clean. The aim of the present paper is to examine some of the results and questions of [3], [30] and [7] from a geometric perspective, in the spirit of [12]. Our thesis is that much of this material is c...
The problem of bounding the “complexity” of a polynomial ideal in terms of the degrees of its gen... more The problem of bounding the “complexity” of a polynomial ideal in terms of the degrees of its generators has attracted a great deal of interest in recent years. Results in this direction go back at least as far as the classical work [16] of Hermann on the ideal membership problem, and the effective Nullstellensatz of Brownawell [4] and Kollár [20] marks a major recent advance. With the development of computational algebraic geometry the question has taken on increasing importance, and it came into particularly clear focus through the influential paper [3] of Bayer and Mumford. More recently, the theorem of [7] and [19] concerning regularity of powers raises the question of bounding the complexity of powers of an ideal, and suggests that asymptotically the picture should become very clean.
Griffiths and Harris showed in 1978 that if E is a rank n vector bundle on a smooth projective va... more Griffiths and Harris showed in 1978 that if E is a rank n vector bundle on a smooth projective variety of dimension n, and if s is a section of E vanishing simply on a finite set Z, then any section of (K_X + det E) vanishing at all but one of the points of Z must also vanish on the remaining one. This generalizes the classical theorem of Cayley-Bacharach, which appears when E is a direct sum of line bundles on projective space. In a recent paper, Mu-Lin Li proposed an extension allowing for the possibility that the zero-locus of s has positive dimensional components, but his result requires a splitting hypothesis that in practice is rarely satisfied. We show that multiplier ideals lead to a quite clean statement in the case of excess vanishing. Along the way, we give some analogues, that seem to be new, of Cayley-Bacharach for finite determinantal loci.
⊆ S the ideal that the polynomials span. Suppose now that J is primary for the irrelevant maximal... more ⊆ S the ideal that the polynomials span. Suppose now that J is primary for the irrelevant maximal ideal m = (x0, . . . , xr), or equivalently that dimC S/J < ∞. In this case J contains all monomials of sufficiently large degree, and it is a classical theorem of Macaulay [5, Theorem 7.4.1] that (1) Jt = St for t ≥ d0 + . . .+ dr − r. Moreover this bound is (always) sharp when p = r. Although less well known, a similar statement holds for powers of J : (2) (J)t = St for t ≥ ad0 + d1 + . . .+ dr − r. This again is always sharp when p = r.
A VANISHING THEOREM FOR LOG CANONICAL PAIRS By TOMMASO DE FERNEX and LAWRENCE EIN
Using inversion of adjunction, we deduce from Nadel’s theorem a vanishing property for ideals she... more Using inversion of adjunction, we deduce from Nadel’s theorem a vanishing property for ideals sheaves on projective varieties, a special case of which recovers a result due to Bertram– Ein–Lazarsfeld. This enables us to generalize to a large class of projective schemes certain bounds on Castelnuovo–Mumford regularity previously obtained by Bertram–Ein–Lazarsfeld in the smooth case and by Chardin–Ulrich for locally complete intersection varieties with rational singularities. Our results are tested on several examples.
In this paper, we show that for a nonsingular projective curve and a positive integer \begin{docu... more In this paper, we show that for a nonsingular projective curve and a positive integer \begin{document}$ k $\end{document}, the \begin{document}$ k $\end{document}-th secant bundle is the blowup of the \begin{document}$ k $\end{document}-th secant variety along the \begin{document}$ (k-1) $\end{document}-th secant variety. This answers a question raised in the recent paper of the authors on secant varieties of curves.
In recent years, the equations defining secant varieties and their syzygies have attracted consid... more In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus g is greater than 2g + 2k + p for nonnegative integers k and p, then the k-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen-Macaulay, and satisfies the property N k+2,p. In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo-Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.
Proceedings of Symposia in Pure Mathematics, Jun 1, 2018
1 There are actually three possible meanings for the statement that a projective variety X ⊆ P r ... more 1 There are actually three possible meanings for the statement that a projective variety X ⊆ P r is cut out by quadrics. The weakest is to ask that this simply be true set-theoretically. A more substantial condition is that X be defined as a subscheme of P r by equations of degree two, ie that the twisted ideal sheaf I X/P r (2) be globally generated. The strongest possibility is that the homogeneous ideal I X of X is generated by elements of degree two. All of the results described here hold in this last sense, although this isn't always the setting in which they were originally established.
If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow t... more 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.
We give a criterion for the vanishing of the weight-one syzygies associated to a line bundle B in... more We give a criterion for the vanishing of the weight-one syzygies associated to a line bundle B in a sufficiently positive embedding of a smooth complex projective variety of arbitrary dimension.
This is a preliminary draft of monograph. It builds on lectures notes on a course that Lawrence E... more This is a preliminary draft of monograph. It builds on lectures notes on a course that Lawrence Ein gave at the University of Catania in Summer 1998, and later again at Hong Kong University in Fall 1999, on lecture notes from the courses that Tommaso de Fernex taught at the University of Utah in Fall 2006, Spring 2010, and Spring 2012, and on lecture notes for the courses taught by Mircea Mustaţȃ in Winter and Fall 2013 at University of Michigan. This draft has been typeset using an edited version of the Springer Monograph class svmono.cls.
We consider pairs (X, A), where X is a variety with klt singularities and A is a formal product o... more We consider pairs (X, A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also assume that X has (formally) bounded singularities, in the sense that it is, formally locally, a subvariety in a fixed affine space defined by equations of bounded degree. We prove in this context a conjecture of Shokurov, predicting that the set of log canonical thresholds for such pairs satisfies the Ascending Chain Condition.
Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimensi... more Shokurov conjectured that the set of all log canonical thresholds on varieties of bounded dimension satisfies the ascending chain condition. In this paper we prove that the conjecture holds for log canonical thresholds on smooth varieties and, more generally, on locally complete intersection varieties and on varieties with quotient singularities.
Using inversion of adjunction, we deduce from Nadel's theorem a vanishing property for ideals she... more Using inversion of adjunction, we deduce from Nadel's theorem a vanishing property for ideals sheaves on projective varieties, a special case of which recovers a result due to Bertram-Ein-Lazarsfeld. This enables us to generalize to a large class of projective schemes certain bounds on Castelnuovo-Mumford regularity previously obtained by Bertram-Ein-Lazarsfeld in the smooth case and by Chardin-Ulrich for locally complete intersection varieties with rational singularities. Our results are tested on several examples.
As is well known, the "usual discrepancy" is defined for a normal Q-Gorenstein variety. By using ... more As is well known, the "usual discrepancy" is defined for a normal Q-Gorenstein variety. By using this discrepancy we can define a canonical singularity and a log canonical singularity. In the same way, by using a new notion, Mather-Jacobian discrepancy introduced in recent papers we can define a "canonical singularity" and a "log canonical singularity" for not necessarily normal or Q-Gorenstein varieties. In this paper, we show basic properties of these singularities, behavior of these singularities under deformations and determine all these singularities of dimension up to 2.
The problem of bounding the “complexity" of a polynomial ideal in terms of the degrees of it... more The problem of bounding the “complexity" of a polynomial ideal in terms of the degrees of its generators has attracted a great deal of interest in recent years. Results in this direction go back at least as far as the classical work [17] of Hermann on the ideal membership problem, and the effective Nullstellensatz of Brownawell [4] and Koll ár [21] marks amajor recent advance.With the development of computational algebraic geometry the question has taken on increasing importance, and it came into particularly clear focus through the influential paper [3] of Bayer andMumford.More recently, the theoremof [8] and [20] concerning regularity of powers raises the question of bounding the complexity of powers of an ideal, and suggests that asymptotically the picture should become very clean. The aim of the present paper is to examine some of the results and questions of [3], [30] and [7] from a geometric perspective, in the spirit of [12]. Our thesis is that much of this material is c...
The problem of bounding the “complexity” of a polynomial ideal in terms of the degrees of its gen... more The problem of bounding the “complexity” of a polynomial ideal in terms of the degrees of its generators has attracted a great deal of interest in recent years. Results in this direction go back at least as far as the classical work [16] of Hermann on the ideal membership problem, and the effective Nullstellensatz of Brownawell [4] and Kollár [20] marks a major recent advance. With the development of computational algebraic geometry the question has taken on increasing importance, and it came into particularly clear focus through the influential paper [3] of Bayer and Mumford. More recently, the theorem of [7] and [19] concerning regularity of powers raises the question of bounding the complexity of powers of an ideal, and suggests that asymptotically the picture should become very clean.
Griffiths and Harris showed in 1978 that if E is a rank n vector bundle on a smooth projective va... more Griffiths and Harris showed in 1978 that if E is a rank n vector bundle on a smooth projective variety of dimension n, and if s is a section of E vanishing simply on a finite set Z, then any section of (K_X + det E) vanishing at all but one of the points of Z must also vanish on the remaining one. This generalizes the classical theorem of Cayley-Bacharach, which appears when E is a direct sum of line bundles on projective space. In a recent paper, Mu-Lin Li proposed an extension allowing for the possibility that the zero-locus of s has positive dimensional components, but his result requires a splitting hypothesis that in practice is rarely satisfied. We show that multiplier ideals lead to a quite clean statement in the case of excess vanishing. Along the way, we give some analogues, that seem to be new, of Cayley-Bacharach for finite determinantal loci.
⊆ S the ideal that the polynomials span. Suppose now that J is primary for the irrelevant maximal... more ⊆ S the ideal that the polynomials span. Suppose now that J is primary for the irrelevant maximal ideal m = (x0, . . . , xr), or equivalently that dimC S/J < ∞. In this case J contains all monomials of sufficiently large degree, and it is a classical theorem of Macaulay [5, Theorem 7.4.1] that (1) Jt = St for t ≥ d0 + . . .+ dr − r. Moreover this bound is (always) sharp when p = r. Although less well known, a similar statement holds for powers of J : (2) (J)t = St for t ≥ ad0 + d1 + . . .+ dr − r. This again is always sharp when p = r.
A VANISHING THEOREM FOR LOG CANONICAL PAIRS By TOMMASO DE FERNEX and LAWRENCE EIN
Using inversion of adjunction, we deduce from Nadel’s theorem a vanishing property for ideals she... more Using inversion of adjunction, we deduce from Nadel’s theorem a vanishing property for ideals sheaves on projective varieties, a special case of which recovers a result due to Bertram– Ein–Lazarsfeld. This enables us to generalize to a large class of projective schemes certain bounds on Castelnuovo–Mumford regularity previously obtained by Bertram–Ein–Lazarsfeld in the smooth case and by Chardin–Ulrich for locally complete intersection varieties with rational singularities. Our results are tested on several examples.
In this paper, we show that for a nonsingular projective curve and a positive integer \begin{docu... more In this paper, we show that for a nonsingular projective curve and a positive integer \begin{document}$ k $\end{document}, the \begin{document}$ k $\end{document}-th secant bundle is the blowup of the \begin{document}$ k $\end{document}-th secant variety along the \begin{document}$ (k-1) $\end{document}-th secant variety. This answers a question raised in the recent paper of the authors on secant varieties of curves.
In recent years, the equations defining secant varieties and their syzygies have attracted consid... more In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus g is greater than 2g + 2k + p for nonnegative integers k and p, then the k-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen-Macaulay, and satisfies the property N k+2,p. In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo-Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.
Proceedings of Symposia in Pure Mathematics, Jun 1, 2018
1 There are actually three possible meanings for the statement that a projective variety X ⊆ P r ... more 1 There are actually three possible meanings for the statement that a projective variety X ⊆ P r is cut out by quadrics. The weakest is to ask that this simply be true set-theoretically. A more substantial condition is that X be defined as a subscheme of P r by equations of degree two, ie that the twisted ideal sheaf I X/P r (2) be globally generated. The strongest possibility is that the homogeneous ideal I X of X is generated by elements of degree two. All of the results described here hold in this last sense, although this isn't always the setting in which they were originally established.
If a morphism of germs of schemes induces isomorphisms of all local jet schemes, does it follow t... more 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.
We give a criterion for the vanishing of the weight-one syzygies associated to a line bundle B in... more We give a criterion for the vanishing of the weight-one syzygies associated to a line bundle B in a sufficiently positive embedding of a smooth complex projective variety of arbitrary dimension.
Uploads
Papers by Lawrence Ein