Let
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 NSFPAR ID:
 10501872
 Publisher / Repository:
 Proceedings of the American Mathematical Society
 Date Published:
 Journal Name:
 Proceedings of the American Mathematical Society
 Volume:
 151
 Issue:
 772
 ISSN:
 00029939
 Page Range / eLocation ID:
 4099 to 4112
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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$R$ be a standard graded algebra over a field. We investigate how the singularities of$\operatorname {Spec} R$ or$\operatorname {Proj} R$ affect the$h$ vector of$R$ , which is the coefficient of the numerator of its Hilbert series. The most concrete consequence of our work asserts that if$R$ satisfies Serre’s condition$(S_r)$ and has reasonable singularities (Du Bois on the punctured spectrum or$F$ pure), then$h_0$ , …,$h_r\geq 0$ . Furthermore the multiplicity of$R$ is at least$h_0+h_1+\dots +h_{r1}$ . We also prove that equality in many cases forces$R$ to be CohenMacaulay. The main technical tools are sharp bounds on regularity of certain$\operatorname {Ext}$ modules, which can be viewed as Kodairatype vanishing statements for Du Bois and$F$ pure singularities. Many corollaries are deduced, for instance that nice singularities of small codimension must be CohenMacaulay. Our results build on and extend previous work by de FernexEin, EisenbudGoto, HunekeSmith, MuraiTerai and others. 
In this paper we consider which families of finite simple groups
$G$ have the property that for each$\epsilon > 0$ there exists$N > 0$ such that, if$G \ge N$ and$S, T$ are normal subsets of$G$ with at least$\epsilon G$ elements each, then every nontrivial element of$G$ is the product of an element of$S$ and an element of$T$ .We show that this holds in a strong and effective sense for finite simple groups of Lie type of bounded rank, while it does not hold for alternating groups or groups of the form
$\mathrm {PSL}_n(q)$ where$q$ is fixed and$n\to \infty$ . However, in the case$S=T$ and$G$ alternating this holds with an explicit bound on$N$ in terms of$\epsilon$ .Related problems and applications are also discussed. In particular we show that, if
$w_1, w_2$ are nontrivial words,$G$ is a finite simple group of Lie type of bounded rank, and for$g \in G$ ,$P_{w_1(G),w_2(G)}(g)$ denotes the probability that$g_1g_2 = g$ where$g_i \in w_i(G)$ are chosen uniformly and independently, then, as$G \to \infty$ , the distribution$P_{w_1(G),w_2(G)}$ tends to the uniform distribution on$G$ with respect to the$L^{\infty }$ norm. 
Let
$f: X \to Y$ be a regular covering of a surface$Y$ of finite type with nonempty boundary, with finitelygenerated (possibly infinite) deck group$G$ . We give necessary and sufficient conditions for an integral homology class on$X$ to admit a representative as a connected component of the preimage of a nonseparating simple closed curve on$Y$ , possibly after passing to a “stabilization”, i.e. a$G$ equivariant embedding of covering spaces$X \hookrightarrow X^+$ . 
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$I$ is an ideal in a Gorenstein ring$S$ , and$S/I$ is CohenMacaulay, then the same is true for any linked ideal$I’$ ; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal$L_{n}$ of minors of a generic$2 \times n$ matrix when$n>3$ .In this paper we initiate the study of a different sort of CohenMacaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of
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