In this paper, we show that Severi varieties parameterizing irreducible reduced planar curves of a given degree and geometric genus are either empty or irreducible in any characteristic. Following Severi's original idea, this gives a new proof of the irreducibility of the moduli space of smooth projective curves of a given genus in positive characteristic. It is the first proof that involves no reduction to the characteristic zero case. As a further consequence, we generalize Zariski's theorem to positive characteristic and show that a general reduced planar curve of a given geometric genus is nodal.
In this paper we show that the derived category of Brauer-Severi curves satisfies the Jordan-Hölder property and cannot have quasi-phantoms, phantoms or universal phantoms. In this way we obtain that quasi-phantoms, phantoms or universal phantoms cannot exist in the derived category of smooth projective curves over a field $k$. Moreover, we show that a $n$-dimensional Brauer-Severi variety is completely characterized by the existence of a full weak exceptional collection consisting of pure vector bundles of length $n+1$, at least in characteristic zero. We conjecture that Brauer-Severi varieti…
Let $X$ be a minimal surface of general type over an algebraically closed field $\mathbf{k}$ of $\mathrm{char}.(\mathbf{k})=p\ge 0$. If the Albanese morphism $a_X:X\to \mathrm{Alb}_X$ is generically finite onto its image, we formulate a constant $c(X,L)\ge 0$ for a very ample line bundle $L$ on $\mathrm{Alb}_X$ such that $c(X,L)=0$ if and only if $\dim \mathrm{Alb}_X=2$ and $a_X: X\to \mathrm{Alb}_X$ is a double cover. A refined Severi inequality $$K^2_X\ge (4+{\rm min}\{\,c(X,L),\,\frac{1}{3}\,\})χ(\mathcal{O}_X)$$ is proved. Then we prove that $K^2_X=4χ(\mathcal{O}_X)$ if and only if the can…
Let $X$ be a surface of general type with maximal Albanese dimension over an algebraically closed field of characteristic greater than two: we prove that if $K_X^2<\frac{9}{2}χ(\mathcal{O}_X)$, one has $K_X^2\geq 4χ(\mathcal{O}_X)+4(q-2)$. Moreover we give a complete classification of surfaces for which equality holds for $q(X)\geq 3$: these are surfaces whose canonical model is a double cover of a product elliptic surface branched over an ample divisor with at most negligible singularities which intersects the elliptic fibre twice. In addition we expose a similar partial result over algebraic…
André and Maulik--Poonen proved that for any smooth proper family $X\to B$ of varieties over an algebraically closed field of characteristic $0$, there is a closed fiber whose Néron-Severi group has the same rank as that of the Néron-Severi group of the geometric generic fiber. We prove the analogous statement over algebraically closed fields of characteristic $p>0$ which are not isomorphic to $\overline{\mathbb{F}}_p$. Furthermore, we prove that for any algebraically closed field $k$ of characteristic $p>0$ and smooth proper family $X\to B$ of $k$-varieties, there exists a dense open subvarie…
For a K3 surface of finite height over a field of odd characteristic, there exists a smooth lifting to the ring of Witt vectors such that the reduction map from the Picard group of the generic fiber to the Picard group of the special fiber is isomorphic. In this paper, using this result, we give a criterion for a K3 surface of finite height over a field of odd characteristic to be an Enriques K3 surface or to be a K3 surface in terms of the Neron-Severi lattice. Then we show every Kummer surface has an Enriques involution. We also give a classification of K3 surfaces of Picard rank of 20 over …
Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety $X$ of dimension $n \ge 2$ to a curve $B$ defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n! χ_f$, which was conjectured by Barja in [2]. Via the strategy outlined in [5], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $χ_f > 0$, we prove that the general fiber $F$ of $f$ has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! χ(F, ω_F)$. We also prove some …
Ardila and Block used tropical results of Brugalle and Mikhalkin to count nodal curves on a certain family of toric surfaces. Building on a linearity result of the first author, we revisit their work in the context of the Goettsche-Yau-Zaslow formula for counting nodal curves on arbitrary smooth surfaces, addressing several questions they raised by proving stronger versions of their main theorems. In the process, we give new combinatorial formulas for the coefficients arising in the Goettsche-Yau-Zaslow formulas, and give correction terms arising from rational double points in the relevant fam…
We study parameter spaces of linear series on projective curves in the presence of unibranch singularities, i.e. {\it cusps}; and to do so, we stratify cusps according to value semigroup. We show that {\it generalized Severi varieties} of maps $\mathbb{P}^1 \rightarrow \mathbb{P}^n$ with images of fixed degree and arithmetic genus are often {\it reducible} whenever $n \geq 3$. We also prove that the Severi variety of degree-$d$ maps with a hyperelliptic cusp of delta-invariant $g \ll d$ is of codimension at least $(n-1)g$ inside the space of degree-$d$ holomorphic maps $\mathbb{P}^1 \rightarro…
We extend to arbitrary characteristic some known results about automorphisms of complex Enriques surfaces that act trivially on the cohomology or the cohomology modulo torsion.
This paper summarizes the results at the present moment about singularities with respect to the Mather-Jacobian log discrepancies over algebraically closed field of arbitrary characteristic. The basic point is the Inversion of Adjunction with respect to Mather-Jacobian discrepancies holds in arbitrary characteristic. Based on this fact we will reduce many geometric properties of the singularities into the problem on jet schemes and try to avoid discussions which are distinctive for characteristic 0.
In the current paper we show that the dimension of a family $V$ of irreducible reduced curves in a given ample linear system on a toric surface $S$ over an algebraically closed field is bounded from above by $-K_S.C+p_g(C)-1$, where $C$ denotes a general curve in the family. This result generalizes a famous theorem of Zariski to the case of positive characteristic. We also explore new phenomena that occur in positive characteristic: We show that the equality $\dim(V)=-K_S.C+p_g(C)-1$ does not imply the nodality of $C$ even if $C$ belongs to the smooth locus of $S$, and construct reducible Seve…
We generalize the recent work of S. Fomin and G. Mikhalkin on polynomial formulas for Severi degrees. The degree of the Severi variety of plane curves of degree d and delta nodes is given by a polynomial in d, provided delta is fixed and d is large enough. We extend this result to generalized Severi varieties parametrizing plane curves which, in addition, satisfy tangency conditions of given orders with respect to a given line. We show that the degrees of these varieties, appropriately rescaled, are given by a combinatorially defined "relative node polynomial" in the tangency orders, provided …
We study the classification of plane curve singularities in arbitrary characteristic. We first give a bound for the determinacy of a plane curve singularity with respect to pararametrization equivalence in terms of its conductor. Then we classify parametrization simple plane curve singularities which are irreducible by giving a concrete list of normal forms of equations and parametrizations. In characteristic zero, the classification of parametrization simple irreducible plane curve singularities was achieved by Bruce and Gaffney.
Let $\Bbbk$ be a field of characteristic zero and $G$ be a finite group of automorphisms of projective plane over $\Bbbk$. Castelnuovo's criterion implies that the quotient of projective plane by $G$ is rational if the field $\Bbbk$ is algebraically closed. In this paper we prove that $\mathbb{P}^2_{\Bbbk} / G$ is rational for an arbitrary field $\Bbbk$ of characteristic zero.
In this paper we characterize two-dimensional semi-log canonical hypersurfaces in arbitrary characteristic from the viewpoint of the initial term of the defining equation. As an application, we prove a conjecture about a uniform bound of divisors computing minimal log discrepancies for two dimensional varieties, which is a conjecture by Ishii and also a special case of the conjecture by Mustaţǎ-Nakamura.
The concept of the maximal contact is the key in Hironaka's resolution theory. It treats local theory, and it is not effective in positive characteristics. This is the essential reason why Hironaka's theory treats only the case of characteristic zero. In this article we propose the substitute for the maximal contact, which is effective in any characteristics of the ground field. We replace the maximal contact by a theorem in the theory of torus embeddings. Using essential ideas here, we would like to establish the theory of resolution of singularities in arbitrary characteristics in a global s…
Over $\C$, Henry Laufer classified all taut surface singularities. We adapt and extent his transcendental methods to positive characteristic. With this we show that if a normal surface singularity is taut over $\C$, then the normal surface singularities with isomorphic dual graph over algebraically closed fields of characteristic exponent $p>1$ are taut for all but finitely many $p$. We conjecture that this is actually "if and only if".
We give an elementary construction of an arbitrary differentially closed field and of a universal differential extension of a differential field in terms of Nash function fields. We also give a characterization of any Archimedean ordered differentially closed field in terms of Nash functions.
Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically closed field of any characteristic. On the other hand, a rank $2$ vector bundle on $E$ defines a surface $S$ with a $\mathbb{P}^1$-bundle structure on $E$. We study when $S$ has an elliptic fibration according to the Atiyah's classification, and what kinds of singular fibers appear.
For quantum Hamiltonian reductions in arbitrary characteristics, it is known that derived localization holds if and only if the algebra of global sections has finite global dimension. In this paper we provide an alternative characterization of when derived localization holds: Derived localization holds if and only if it holds for an explicit finite set of (quantized) line bundles. As an application, we prove a new result that there are integral weights for which localization holds on in the positive characteristic hypertoric case for $p$ larger than an explicit bound. We also discuss how deriv…
Let $A$ be an abelian variety over an algebraically closed field. We show that $A$ is the automorphism group scheme of some smooth projective variety if and only if $A$ has only finitely many automorphisms as an algebraic group. This generalizes a result of Lombardo and Maffei for complex abelian varieties.
The goal of this paper is to prove the full geometric Bogomolov conjecture. We first reduce it to the case that the extension of the base fields has transcendence degree 1, and then we prove the later case by intersection theory in algebraic geometry. The proof uses Yamaki's reduction theorem on the geometric Bogomolov conjecture and the Manin--Mumford conjecture proved by Raynaud and Hrushovski.
It is known that Chern characteristic numbers of compact complex manifolds cannot have arbitrary values. They satisfy certain divisability conditions. W. Ebeling and S. M. Gusein-Zade gave a definition of Chern characteristic numbers of singular compact complex analytic varieties. We prove that there exists a singular projective variety with an arbitrary set of characteristic numbers.
We prove that there exists a supersingular nonsingular curve of genus $4$ in arbitrary characteristic $p$. For $p>3$ we shall prove that the desingularization of a certain fiber product over $\mathbb{P}^1$ of two supersingular elliptic curves is supersingular.
We prove a conjecture of Milne pertaining to the existence of integral canonical models of Shimura varieties of abelian type in arbitrary unramified mixed characteristic $(0,p)$. As an application we prove for $p=2$ a motivic conjecture of Milne pertaining to integral canonical models of Shimura varieties of Hodge type.
We extend the recent classification of Hilbert schemes with two Borel-fixed points to arbitrary characteristic. We accomplish this by synthesizing Reeves' algorithm for generating strongly stable ideals with the basic properties of Borel-fixed ideals and our previous work classifying Hilbert schemes with unique Borel-fixed points.
We prove a theorem of Tits type for automorphism groups of projective varieties over an algebraically closed field of arbitrary characteristic, which was first conjectured by Keum, Oguiso and Zhang for complex projective varieties.
We show an analogue of Jordan's theorem for algebraic groups defined over a field $\mathbb k$ of arbitrary characteristic. As a consequence, a Jordan-type property holds for the automorphism group of any projective variety over $\mathbb k$.
We prove a certain 'fat hyperplane section' Weak Lefschetz-type theorem for etale cohomology of non-projective varieties, similar to a result of Goresky and MacPherson (over complex numbers). This statement easily yields certain (vast) generalizations of the 'ordinary' Weak Lefschetz and Barth's theorems in arbitrary characteristic (that do not require any stratified Morse theory for their proof).
In this paper we consider Grassmannians in arbitrary characteristic. Generalizing Kapranov's well-known characteristic-zero results we construct dual exceptional collections on them (which are however not strong) as well as a tilting bundle. We show that this tilting bundle has a quasi-hereditary endomorphism ring and we identify the standard, costandard, projective and simple modules of the latter.
It is well-known that theta characteristics on smooth plane curves over a field of characteristic different from two are in bijection with certain smooth complete intersections of three quadrics. We generalize this bijection to possibly singular hypersurfaces of any dimension over arbitrary fields including those of characteristic two. It is accomplished in terms of linear orbits of tuples of symmetric matrices instead of smooth complete intersections of quadrics. As an application of our methods, we give a description of the projective automorphism groups of complete intersections of quadrics…
Mustaţă defined Bernstein-Sato polynomials in prime characteristic for principal ideals and proved that the roots of these polynomials are related to the $F$-jumping numbers of the ideal. This approach was later refined by Bitoun. Here we generalize these techniques to develop analogous notions for the case of arbitrary ideals and prove that these have similar connections to $F$-jumping numbers.
Let $X$ be an integral affine or projective scheme of finite presentation over a perfect field. We prove that $X$ admits a resolution, that is, there exists a smooth scheme $\widetilde X$ and a projective birational morphism from $\widetilde X$ onto $X$.
We show that for any integer n and any field k of characteristic different from 2 there are at most finitely many isomorphism classes of quadratic morphisms from the projective line over k to itself with a finite postcritical orbit of size n. As a consequence we prove that every postcritically finite quadratic morphism over a field of positive characteristic can be lifted to characteristic zero with the same combinatorial type of postcritical orbit. The associated profinite geometric monodromy group is therefore the same as in characteristic zero, where it can be described explicitly by genera…
We explicitly find an equation and a projective embedding of the Kummer surface associated to the Jacobian of a curve of genus 2 given by an equation of the form y^2 + h(x)y = f(x) over an arbitrary ground field as well as several maps that can be used to perform arithmetic on it. This extends earlier work by Flynn and has applications, for instance, in computations of canonical heights for genus 2 Jacobians and in cryptography.
