In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying
 Award ID(s):
 1802139
 NSFPAR ID:
 10252047
 Date Published:
 Journal Name:
 International Mathematics Research Notices
 ISSN:
 10737928
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract , for any even integer$$K^2 = 4p_g8$$ ${K}^{2}=4{p}_{g}8$ . These surfaces also have unbounded irregularity$$p_g\ge 4$$ ${p}_{g}\ge 4$q . We carry out our study by investigating the deformations of the canonical morphism , where$$\varphi :X\rightarrow {\mathbb {P}}^N$$ $\phi :X\to {P}^{N}$ is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):54895507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of$$\varphi $$ $\phi $ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of$$\varphi $$ $\phi $ is twotoone onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of$$\varphi $$ $\phi $X are unobstructed even though does not vanish. Consequently,$$H^2(T_X)$$ ${H}^{2}\left({T}_{X}\right)$X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality , with an irreducible component that has a proper quadruple sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with$$p_g > 2q4$$ ${p}_{g}>2q4$ , studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus$$K^2 = 2p_g 4$$ ${K}^{2}=2{p}_{g}4$ , for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.$$g\ge 3$$ $g\ge 3$ 
Abstract Positive
intermediate Ricci curvature on a Riemannian$$k\mathrm{th}$$ $k\mathrm{th}$n manifold, to be denoted by , is a condition that interpolates between positive sectional and positive Ricci curvature (when$${{\,\mathrm{Ric}\,}}_k>0$$ ${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$ and$$k =1$$ $k=1$ respectively). In this work, we produce many examples of manifolds of$$k=n1$$ $k=n1$ with$${{\,\mathrm{Ric}\,}}_k>0$$ ${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$k small by examining symmetric and normal homogeneous spaces, along with certain metric deformations of fat homogeneous bundles. As a consequence, we show that every dimension congruent to$$n\ge 7$$ $n\ge 7$ supports infinitely many closed simply connected manifolds of pairwise distinct homotopy type, all of which admit homogeneous metrics of$$3\,{{\,\mathrm{mod}\,}}4$$ $3\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{mod}\phantom{\rule{0ex}{0ex}}4$ for some$${{\,\mathrm{Ric}\,}}_k>0$$ ${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$ . We also prove that each dimension$$k $k<n/2$ congruent to 0 or$$n\ge 4$$ $n\ge 4$ supports closed manifolds which carry metrics of$$1\,{{\,\mathrm{mod}\,}}4$$ $1\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}\mathrm{mod}\phantom{\rule{0ex}{0ex}}4$ with$${{\,\mathrm{Ric}\,}}_k>0$$ ${\phantom{\rule{0ex}{0ex}}\mathrm{Ric}\phantom{\rule{0ex}{0ex}}}_{k}>0$ , but do not admit metrics of positive sectional curvature.$$k\le n/2$$ $k\le n/2$ 
null (Ed.)Abstract The elliptic algebras in the title are connected graded $\mathbb {C}$ algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{\mathcal {L}_{n/k}}:E^g \to \mathbb {P}^{n1}$ embeds $E^g$ as a projectively normal subvariety that is a schemetheoretic intersection of quadric and cubic hypersurfaces.more » « less

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was measured at BGOOD at ELSA from threshold to a centreofmass energy of$$\gamma n\rightarrow K^0\Sigma ^0$$ $\gamma n\to {K}^{0}{\Sigma}^{0}$ . Close to threshold the results are consistent with existing data and are in agreement with partial wave analysis solutions over the full measured energy range, with a large coupling to the$$2400\,\hbox {MeV}$$ $2400\phantom{\rule{0ex}{0ex}}\text{MeV}$ evident. This is the first dataset covering the$$\Delta (1900)1/2^$$ $\Delta \left(1900\right)1/{2}^{}$ threshold region, where there are model predictions of dynamically generated vector mesonbaryon resonance contributions.$$K^*$$ ${K}^{\ast}$ 
null (Ed.)Abstract. This work measured $ \mathrm{d}\sigma/\mathrm{d}\Omega$ d σ / d Ω for neutral kaon photoproduction reactions from threshold up to a c.m. energy of 1855MeV, focussing specifically on the $ \gamma p\rightarrow K^0\Sigma^+$ γ p → K 0 Σ + , $ \gamma n\rightarrow K^0\Lambda$ γ n → K 0 Λ , and $ \gamma n\rightarrow K^0 \Sigma^0$ γ n → K 0 Σ 0 reactions. Our results for $ \gamma n\rightarrow K^0 \Sigma^0$ γ n → K 0 Σ 0 are the firstever measurements for that reaction. These data will provide insight into the properties of $ N^{\ast}$ N * resonances and, in particular, will lead to an improved knowledge about those states that couple only weakly to the $ \pi N$ π N channel. Integrated cross sections were extracted by fitting the differential cross sections for each reaction as a series of Legendre polynomials and our results are compared with prior experimental results and theoretical predictions.more » « less