In this paper, we develop a technique for discovering (noneffective) irrational rays at the boundary of the Mori cone for linear systems on a general blowup of the plane and give examples of such irrational rays.
Based on the work of Harada, Nowroozi, and Van Tuyl, who provided particular length two virtual resolutions for finite sets of points in F1 & times; F1, we prove that the vast majority of virtual resolutions of a pair for minimal elements of the multigraded regularity in this setting are of Hilbert-Burch type. We give explicit descriptions of these short virtual resolutions that depend only on the number of points. Moreover, despite initial evidence, we show that these virtual resolutions are not always short and give sufficient conditions for their length to be three.
Let X_0 be an irreducible smooth projective curve defined over ℚ and f_0 : X_0 →ℙ^1_ℚ a nonconstant morphism whose branch locus is contained in the subset {0,1, ∞}⊂ℙ^1_ℚ. For any vector bundle E on X = X_0×_ Spec ℚ Specℂ, consider the direct image f_*E on ℙ^1_ℂ, where f= (f_0)_ℂ. It decomposes into a direct sum of line bundles and also it has a natural parabolic structure. We prove that E is the base change, to ℂ, of a vector bundle on X_0 if and only if there is an isomorphism f_*E ∼→⊕_i=1^r 𝒪_ℙ^1_ℂ(m_i), where r = rank(f_*E), that takes the parabolic structure on f_*E to a parabolic structure on ⊕_i=1^r 𝒪_ℙ^1_ℂ(m_i) defined over ℚ.
In we examined the relationship between the singular set of a compact Riemannian orbifold and the spectrum of the Hodge Laplacian on p-forms by computing the heat invariants associated to the p-spectrum. We showed that the heat invariants of the 0-spectrum together with those of the 1-spectrum for the corresponding Hodge Laplacians are sufficient to distinguish orbifolds from manifolds as long as the singular sets have codimension ≤ 3. This is enough to distinguish orbifolds from manifolds for dimension ≤ 3. Here we give both positive and negative inverse spectral results for the individual p-spectra considered separately. For example, we give conditions on the codimension of the singular set which guarantee that the volume of the singular set is determined, and in many cases we show by providing counterexamples that the conditions are sharp.
We prove that among all constant width bodies of revolution, the minimum of the ratio of the volume to the cubed width is attained by the constant width body obtained by rotation of the Reuleaux triangle about an axis of symmetry.
Let d >= 2 and V belong to a reverse H & ouml;lder class. Let u be a strong solution to the Schr & ouml;dinger equation-Delta u +Vu= f in IIgd. For an appropriate Musielak-Orlicz function phi, we show that HD2uHL phi(& centerdot;)(& Ropf;d) +HVuHL phi(& centerdot;)(& Ropf;d) <= CHfHL phi(& centerdot;)(& Ropf;d).
Fix a smooth projective family of curves $C \to S$ and a split reductive group scheme $G$ over a Noetherian base scheme $S$. For any (possibly nonreduced) fixed relative Cartier divisor $D$, we provide a treatment of the moduli of $G$-bundles on the fibers of $C$ equipped with $t$-connections with pole orders bounded by $D$. Under mild assumptions on the characteristics of all the residue fields of $S$, we construct a Hodge moduli space $M_{Hod, G} \to \mathbb{A}^1_S$ for the semistable locus, construct a Harder-Narasimhan stratification, and thus obtain a semistable reduction theorem. If all the fibers of the divisor of poles $D$ are nonempty, then we show that the stack of semistable objects is smooth over $\mathbb{A}^1_{S}$. We also define a Hodge-Hitchin morphism in positive characteristic and prove that it is proper.
A marked strongly invertible knot is a triple (K,h,δ) of a knot K in S^3, a strong inversion h of K, and a subarc δ⊂Fix(h)≅ S^1 bounded by Fix(h)∩ K≅ S^0. An invariant Seifert surface for (K,h,δ) is an h-invariant Seifert surface for K that intersects Fix(h) in the arc δ. In this paper, we completely determine the equivariant genus (the minimum of the genera of invariant Seifert surfaces for (K,h,δ)) of every marked strongly invertible knot (K,h,δ) with K a 2-bridge knot.
We introduce the notion of halfspaces associated to a group splitting, and investigate the relationship between the coarse geometry of the halfspaces and the coarse geometry of the group. Roughly speaking, the halfspaces of a group splitting are subgraphs of the Cayley graph obtained by pulling back the halfspaces of the Bass–Serre tree. Our first theorem shows that (under mild conditions) any splitting of a one-ended group can be upgraded to a splitting where all the halfspaces are one-ended. Our second theorem demonstrates that a one-ended group usually has a JSJ splitting where all the halfspaces are one-ended. And our third theorem states that if a one-ended finitely presented group G admits a splitting such that some edge stabilizer has more than one end, but the halfspaces associated to the edge stabilizer are one-ended, then H^2(G,ℤG){0}; in particular G is not simply connected at infinity and G is not an n-dimensional duality group for n≥3.
We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Plücker formulas are polynomials in the degree, just as the classical Plücker formulas counting the bitangents and flexes of a degree d generic plane curve. We establish an upper bound for the degrees of these polynomials, and we calculate the leading terms of those whose degrees reach this upper bound. We believe that the paper is understandable without detailed knowledge of equivariant cohomology. It may serve as a demonstration of the use of equivariant cohomology in enumerative geometry through the examples of coincident root strata. We also explain how the equivariant method can be "translated" into the traditional non-equivariant method of resolutions.
A line bundle is immaculate if its cohomology vanishes in every dimension. We give a criterion for when a smooth toric Deligne-Mumford stack has infinitely many immaculate line bundles. This answers positively a question of Borisov and Wang.
A primitive multiple scheme is a Cohen-Macaulay scheme $Y$ such that the associated reduced scheme $X=Y_{red}$ is smooth, irreducible, and that $Y$ can be locally embedded in a smooth variety of dimension $\dim(X)+1$. If $I_X$ is the ideal sheaf of $X$ in $Y$ and $Y\not=X$, then $L=I_X/I_X^2$ is a line bundle on $X$, called the associated line bundle of $Y$. Even if $X$ is projective, $Y$ needs not to be quasi projective. We define in every case the reduced Hilbert polynomial $P_{red,O_X(1)}(E)$ of a coherent sheaf $E$ on $Y$, depending on the choice of an ample line bundle $O_X(1)$ on $X$. If $E$ is a flat family of sheaves on $Y$ parameterized by a smooth curve $C$, then $P_{red,O_X(1)}(E_c)$ does not depend on $c\in C$. We study flat families of sheaves in two important cases: the families of quasi locally free sheaves, and if $n=2$ those of balanced sheaves. Balanced sheaves are generalizations of vector bundles on $Y$, and could be used to expand already known moduli spaces of vector bundles on $Y$. When $X$ is a smooth projective surface, and $Y$ is of multiplicity 2 we study the simplest examples of balanced sheaves: the sheaves $E$ such that there is an exact sequence \[0\longrightarrow I_P\otimes L\longrightarrow E\longrightarrow I_P=E_{|X} \longrightarrow 0 \ , \] where $I_P\subset O_X$ is the ideal sheaf of a point $P\in X$. They can also be described as the ideal sheaves $E$ of subschemes of $Y$ concentrated on $P$, and such that $E_P$ is generated by two elements whose images in $O_{X,P}$ generate the maximal ideal. There is a moduli space for such sheaves, which is an affine bundle on $X$ with associated vector bundle $T_X\otimes L$ (where $T_X$ is the tangent bundle of $X$). The associated class in $H^1(X,T_X\otimes L)$ can be determined.
We prove an asymptotic formula for the number of d-fold partition diamonds of n and their Schmidt-type counterparts. In order to do so, we study the asymptotic behavior of certain infinite products. We also remark on interesting potential connections with mathematical physics and Bloch groups.
A Lorenz link is equivalent to a T-link, which is a positive braid built by concatenating torus braids of increasing size. When each torus braid except the largest is obtained by full twists, then the T-link can be described as the Dehn filling of a parent link. In this paper, we completely classify when such parent links are hyperbolic. This gives a classification of the geometry of T-links obtained by full twists when the amount of twisting is large, although the bound on the number of required twists is not effective. We also present effective results on hyperbolicity for two families of T-links obtained by twisting, even when the number of twists is small. Finally, we identify families of satellite T-links obtained by half-twists.
We study the restriction of Brill-Noether loci to the gonality stratification of the moduli space of curves of fixed genus. As an application, we give new proofs that Brill-Noether loci with ρ=-1,-2 have distinct support, and for fixed r give lower bounds on when one direction of the non-containments of the Maximal Brill-Noether Loci Conjecture hold for Brill-Noether loci of rank r linear systems. Using these techniques, we also show that Brill-Noether loci corresponding to rank 2 linear systems are maximal as soon as g≥ 28 and prove the Maximal Brill-Noether Loci Conjecture for g=20.
We use presentations of the Cox rings of projectivized toric vector bundles and elements of matroid theory to compute Newton-Okounkov bodies, effective cones, and nef cones of these spaces. As an application we analyze the Fano property and establish Fujita's freeness and ampleness conjectures for several classes of projectivized toric vector bundles.
For any prime $p$ and $\varepsilon>0$ we prove that for any sufficiently large positive odd integer $s$ at least $(c_p-\varepsilon) \sqrt{\frac{s}{\log s}}$ of the $p$-adic zeta values $\zeta_p(3),\zeta_p(5),\dots,\zeta_p(s)$ are irrational. The constant $c_p$ is positive and does only depend on $p$. This result establishes a $p$-adic version of the elimination technique used by Fischler--Sprang--Zudilin and Lai--Yu to prove a similar result on classical zeta values. The main difficulty consists in proving the non-vanishing of the resulting linear forms. We overcome this problem by using a new irrationality criterion.
