A construction of complex analytic elliptic cohomology from double free loop spaces

We construct a complex analytic version of an equivariant cohomology theory which appeared in a paper of Rezk, and which is roughly modelled on the Borel-equivariant cohomology of the double free loop space. The construction is defined on finite, torus-equivariant CW complexes and takes values in coherent holomorphic sheaves over the moduli stack of complex elliptic curves. Our methods involve an inverse limit construction over all finite-dimensional subcomplexes of the double free loop space, following an analogous construction of Kitchloo for single free loop spaces. We show that, for any given complex elliptic curve $\mathcal {C}$ , the fiber of our construction over $\mathcal {C}$ is isomorphic to Grojnowski's equivariant elliptic cohomology theory associated to $\mathcal {C}$ .

K3 curves with index $$k>1$$

Let $$\mathcal {KC}_g ^k$$ KC g k be the moduli stack of pairs (S, C) with S a K3 surface and $$C\subseteq S$$ C ⊆ S a genus g curve with divisibility k in $$\mathrm {Pic}(S)$$ Pic ( S ) . In this article we study the forgetful map $$c_g^k:(S,C) \mapsto C$$ c g k : ( S , C ) ↦ C from $$\mathcal {KC}_g ^k$$ KC g k to $${\mathcal {M}}_g$$ M g for $$k>1$$ k > 1 . First we compute by geometric means the dimension of its general fibre. This turns out to be interesting only when S is a complete intersection or a section of a Mukai variety. In the former case we find the existence of interesting Fano varieties extending C in its canonical embedding. In the latter case this is related to delicate modular properties of the Mukai varieties. Next we investigate whether $$c_g^k$$ c g k dominates the locus in $${\mathcal {M}}_g$$ M g of k-spin curves with the appropriate number of independent sections. We are able to do this only when S is a complete intersection, and obtain in these cases some classification results for spin curves.

Smooth covers of moduli stacks of Riemann surfaces with symmetry

We construct explicitly a finite cover of the moduli stack of compact Riemann surfaces with a given group of symmetries by a smooth quasi-projective variety.

Motivic Hypercohomology Solutions in Field Theory and Applications in H-States

Triangulated derived categories are considered which establish a commutative scheme (triangle) for determine or compute a hypercohomology of motives for the obtaining of solutions of the field equations. The determination of this hypercohomology arises of the derived category $\textup{DM}_{\textup {gm}}(k)$,  which is of the motivic objects whose image is under $\textup {Spec}(k)$  that is to say, an equivalence of the underlying triangulated tensor categories, compatible with respective functors on $\textup{Sm}_{k}^{\textup{Op}}$. The geometrical motives will be risked with the moduli stack to holomorphic bundles. Likewise, is analysed the special case where complexes $C=\mathbb{Q}(q)$,  are obtained when cohomology groups of the isomorphism $H_{\acute{e}t}^{p}(X,F_{\acute{e}t})\cong (X,F_{Nis})$,   can be vanished for  $p>\textup{dim}(Y)$.  We observe also the Beilinson-Soul$\acute{e}$ vanishing  conjectures where we have the vanishing $H^{p}(F,\mathbb{Q}(q))=0, \ \ \textup{if} \ \ p\leq0,$ and $q>0$,   which confirms the before established. Then survives a hypercohomology $\mathbb{H}^{q}(X,\mathbb{Q})$. Then its objects are in $\textup{Spec(Sm}_{k})$.  Likewise, for the complex Riemannian manifold the integrals of this hypercohomology are those whose functors image will be in $\textup{Spec}_{H}\textup{SymT(OP}_{L_{G}}(D))$, which is the variety of opers on the formal disk $D$, or neighborhood of all point in a surface $\Sigma$.  Likewise, will be proved that $\mathrm{H}^{\vee}$,  has the decomposing in components as hyper-cohomology groups which can be characterized as H- states in Vec$_\mathbb{C}$, for field equations $d \textup{da}=0$,  on the general linear group with $k=\mathbb{C}$.  A physics re-interpretation of the superposing, to the dual of the spectrum $\mathrm{H}^{\vee}$,  whose hypercohomology is a quantized version of the cohomology space $H^{q}(Bun_{G},\mathcal{D}^{s})=\mathbb{H}^{q}_{G[[z]]}(\mathrm{G},(\land^{\bullet}[\Sigma^{0}]\otimes \mathbb{V}_{critical},\partial))$ is the corresponding deformed derived category for densities $\mathrm{h} \in \mathrm{H}$, in quantum field theory.

Brauer groups of moduli of hyperelliptic curves via cohomological invariants

Using the theory of cohomological invariants for algebraic stacks, we compute the Brauer group of the moduli stack of hyperelliptic curves ${\mathcal {H}}_g$ over any field of characteristic $0$ . In positive characteristic, we compute the part of the Brauer group whose order is prime to the characteristic of the base field.

Birational models of 𝓜2,2 arising as moduli of curves with nonspecial divisors

We study birational projective models of 𝓜2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜 2,2(𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M 2,2(𝓩).