Singularity Subschemes and Generic Projections

Roberts, Joel

doi:10.2307/1998623

Cited by 6 publications

(5 citation statements)

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“…on p. 473], it suffices to show that for every closed point x ∈ X, the completion O X,x of the local ring at x is F -injective. In [Rob75,(13.2)], Roberts provides a list of possibilities for the isomorphism classes of these complete local rings, which we treat individually:…”

Section: Singularities Of Generic Projection Hypersurfacesmentioning

confidence: 99%

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Permanence properties of $F$-injectivity

Datta¹,

Murayama²

2019

Preprint

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We prove that F -injectivity localizes, descends under faithfully flat homomorphisms, and ascends under flat homomorphisms with Cohen-Macaulay and geometrically F -injective fibers, all for arbitrary Noetherian rings of prime characteristic. As consequences of these results, we show that the F -injective locus is open on rings essentially of finite type over excellent local rings. As a geometric application, we prove that if X is a smooth projective variety of dimension r ≤ 5 over an algebraically closed field of characteristic p > 3 embedded via a d-uple embedding for d ≥ 3r, then every generic projection of X is F -pure, and hence F -injective. This geometric result is the positive characteristic analogue of a theorem of Doherty. Contents 2.1. F -injective and CMFI rings 2.2. The relative Frobenius homomorphism 3. Localization and descent under faithfully flat homomorphisms 3.1. Characterizations of F -injectivity using module-finite algebras 3.2. Localization of F -injectivity 3.3. Descent of F -injectivity 3.4. Geometrically CMFI rings and infinite purely inseparable extensions 4. Ascent under flat homomorphisms with geometrically CMFI fibers 4.1. Preliminaries on local cohomology 4.2. Proof of Theorem A 4.3. Geometrically F -injective rings and finitely generated field extensions 5. CMFI homomorphisms and openness of F -injective loci 5.1. Definition and properties of CMFI homomorphisms 5.2. Openness of the F -injective locus 5.3. F -injectivity of graded rings 6. Singularities of generic projection hypersurfaces Appendix A. F -rationality descends and implies F -injectivity References

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Section: Singularities Of Generic Projection Hypersurfacesmentioning

confidence: 99%

“…The assumption on characteristic is needed to rule out some exceptional cases in the classification in [Rob75,(13.2)], and the d-uple embedding is used to ensure that Y is appropriately embedded in the sense of [Rob75,§9]. Theorem C is the positive characteristic analogue of a theorem of Doherty [Doh08,Main Thm.…”

Section: Introduction Notation Introductionmentioning

confidence: 99%

Permanence properties of $F$-injectivity

Datta¹,

Murayama²

2019

Preprint

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“…points are infinitely near each other, so the theorem yields enumerative formulas for the united ktuples which are the union of Thom-Boardman S^-singularities, i = 1,..., d, qi H h qd = k, cf. Roberts [8].…”

Section: Theorem Given a K-quasi-resolution As Above Assume That Ukmentioning

confidence: 99%

A united-set formula

Ran¹

1983

Bull. Amer. Math. Soc.

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Let a morphism ƒ : X -• Y of algebraic varieties be given. A united set or united k-tuple 2 for ƒ is a fc-tuple x±,..., Xk of distinct points on (or "infinitely near") X, such that /(xi) = ••• = f(xk)> The purpose of this note is to announce an enumerative formula, valid under a restrictive hypothesis, for the united &-tuples of a map, i.e., a formula for the rational equivalence (or homology) class of a suitable cycle which parameterizes them. This yields as special cases formulas for the united /c-tuples which contain a fci-tuple, a /c2-tuple, etc. of mutually infinitely-near points. For our united-A>tuple cycle even to be defined, the morphism ƒ has to admit a certain kind of "resolution" (essentially it must factor through a "generic" map into a variety fibred by smooth curves over Y). Our result is sufficient, however, to yield formulas for the lines having prescribed contacts with a given projective variety having "generic" singularities and arbitrary dimension and codimension; these in turn yield formulas for the Thom-Boardman-Roberts singularity schemes [8] of a generic projection of such a variety. Classically such formulas were known for curves, for surfaces in P 3 , and in a few other cases, cf.[1]. Some recent results were obtained by Lascoux [6], Roberts [9] and LeBarz [7]. Our result yields new formulas even for surfaces in P 4 . For a modern account of these and related matters, see Kleiman's surveys [3, 5].Admittedly, the hypothesis of existence of a "resolution" is a severe restriction on the morphism ƒ. I am hopeful, however, that by pursuing further the same principles as in this paper, I will eventually obtain a united-set formula valid without such a restriction, and which would moreover be completely "intrinsic", in the sense of taking place on a suitable space associated solely to X (which is not the case with the present formula).We shall work in the category of complete (usually nonsingular) varieties over a field. Everything goes through with no change, however, in the category of compact complex manifolds.

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“…Applying a variety of techniques to the results of [Rob75], combined with the above, leads to the following extension of the classical result on curves. …”

Section: Introductionmentioning

confidence: 99%

“…Applying an analogous technique to higher-dimensional varieties seems a natural extension -can we draw any conclusions in this more general setting? Joel Roberts ([Rob71], [Rob75]) provides a useful starting point for understanding the singularities introduced by linearly projecting varieties. Applying the more recent machinery of birational geometry to his work offers the possibility of finding a useful answer to this question.…”

Section: Introductionmentioning

confidence: 99%

Singularities of generic projection hypersurfaces

Doherty

2008

Proc. Amer. Math. Soc.

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Abstract. Linearly projecting smooth projective varieties provide a method of obtaining hypersurfaces birational to the original varieties. We show that in low dimension, the resulting hypersurfaces only have Du Bois singularities. Moreover, we conclude that these Du Bois singularities are in fact semi log canonical. However, we demonstrate the existence of counterexamples in high dimension -the generic linear projection of certain varieties of dimension 30 or higher is neither semi log canonical nor Du Bois.

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Singularity Subschemes and Generic Projections

Cited by 6 publications

References 2 publications

Permanence properties of $F$-injectivity

Permanence properties of $F$-injectivity

A united-set formula

Singularities of generic projection hypersurfaces

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