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The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.
Let
$Y$
be a complex Enriques surface whose universal cover
$X$
is birational to a general quartic Hessian surface. Using the result on the automorphism group of
$X$
due to Dolgachev and Keum, we obtain a finite presentation of the automorphism group of
$Y$
. The list of elliptic fibrations on
$Y$
and the list of combinations of rational double points that can appear on a surface birational to
$Y$
are presented. As an application, a set of generators of the automorphism group of the generic Enriques surface is calculated explicitly.
We give a bound on the primes dividing the denominators of invariants of Picard curves of genus 3 with complex multiplication. Unlike earlier bounds in genus 2 and 3, our bound is based, not on bad reduction of curves, but on a very explicit type of good reduction. This approach simultaneously yields a simplification of the proof and much sharper bounds. In fact, unlike all previous bounds for genus 3, our bound is sharp enough for use in explicit constructions of Picard curves.
The jaggedness of an order ideal
$I$
in a poset
$P$
is the number of maximal elements in
$I$
plus the number of minimal elements of
$P$
not in
$I$
. A probability distribution on the set of order ideals of
$P$
is toggle-symmetric if for every
$p\in P$
, the probability that
$p$
is maximal in
$I$
equals the probability that
$p$
is minimal not in
$I$
. In this paper, we prove a formula for the expected jaggedness of an order ideal of
$P$
under any toggle-symmetric probability distribution when
$P$
is the poset of boxes in a skew Young diagram. Our result extends the main combinatorial theorem of Chan–López–Pflueger–Teixidor [Trans. Amer. Math. Soc., forthcoming. 2015, arXiv:1506.00516], who used an expected jaggedness computation as a key ingredient to prove an algebro-geometric formula; and it has applications to homomesies, in the sense of Propp–Roby, of the antichain cardinality statistic for order ideals in partially ordered sets.
Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over
$\mathbb{Q}$
with good reduction away from 3, up to
$\mathbb{Q}$
-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined and an application to a question of Ihara is discussed.
We prove that the canonical ring of a canonical variety in the sense of de Fernex and Hacon is finitely generated. We prove that canonical varieties are Kawamata log terminal (klt) if and only if is finitely generated. We introduce a notion of nefness for non-ℚ-Gorenstein varieties and study some of its properties. We then focus on these properties for non-ℚ-Gorenstein toric varieties.
We describe the construction of a database of genus-
$2$
curves of small discriminant that includes geometric and arithmetic invariants of each curve, its Jacobian, and the associated
$L$
-function. This data has been incorporated into the
$L$
-Functions and Modular Forms Database (LMFDB).
Given a sextic CM field
$K$
, we give an explicit method for finding all genus-
$3$
hyperelliptic curves defined over
$\mathbb{C}$
whose Jacobians are simple and have complex multiplication by the maximal order of this field, via an approximation of their Rosenhain invariants. Building on the work of Weng [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], we give an algorithm which works in complete generality, for any CM sextic field
$K$
, and computes minimal polynomials of the Rosenhain invariants for any period matrix of the Jacobian. This algorithm can be used to generate genus-3 hyperelliptic curves over a finite field
$\mathbb{F}_{p}$
with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo
$p$
.
We study cup products in the integral cohomology of the Hilbert scheme of
$n$
points on a K3 surface and present a computer program for this purpose. In particular, we deal with the question of which classes can be represented by products of lower degrees.
If
$S$
is a quintic surface in
$\mathbb{P}^{3}$
with singular set 15 3-divisible ordinary cusps, then there is a Galois triple cover
${\it\phi}:X\rightarrow S$
branched only at the cusps such that
$p_{g}(X)=4$
,
$q(X)=0$
,
$K_{X}^{2}=15$
and
${\it\phi}$
is the canonical map of
$X$
. We use computer algebra to search for such quintics having a free action of
$\mathbb{Z}_{5}$
, so that
$X/\mathbb{Z}_{5}$
is a smooth minimal surface of general type with
$p_{g}=0$
and
$K^{2}=3$
. We find two different quintics, one of which is the van der Geer–Zagier quintic; the other is new.
We also construct a quintic threefold passing through the 15 singular lines of the Igusa quartic, with 15 cuspidal lines there. By taking tangent hyperplane sections, we compute quintic surfaces with singular sets
$17\mathsf{A}_{2}$
,
$16\mathsf{A}_{2}$
,
$15\mathsf{A}_{2}+\mathsf{A}_{3}$
and
$15\mathsf{A}_{2}+\mathsf{D}_{4}$
.
In this paper, we investigate examples of good and optimal Drinfeld modular towers of function fields. Surprisingly, the optimality of these towers has not been investigated in full detail in the literature. We also give an algorithmic approach for obtaining explicit defining equations for some of these towers and, in particular, give a new explicit example of an optimal tower over a quadratic finite field.
Mori dream spaces form a large example class of algebraic varieties, comprising the well-known toric varieties. We provide a first software package for the explicit treatment of Mori dream spaces and demonstrate its use by presenting basic sample computations. The software package is accompanied by a Cox ring database which delivers defining data for Cox rings and Mori dream spaces in a suitable format. As an application of the package, we determine the common Cox ring for the symplectic resolutions of a certain quotient singularity investigated by Bellamy–Schedler and Donten-Bury–Wiśniewski.
We show how to efficiently evaluate functions on Jacobian varieties and their quotients. We deduce an algorithm to compute
$(l,l)$
isogenies between Jacobians of genus two curves in quasi-linear time in the degree
$l^{2}$
.
We compute the global log canonical thresholds of quasi-smooth well-formed complete intersection log del Pezzo surfaces of amplitude 1 in weighted projective spaces. As a corollary we show the existence of orbifold Kähler—Einstein metrics on many of them.
We consider higher secant varieties to Veronese varieties. Most points on the rth secant variety are represented by a finite scheme of length r contained in the Veronese variety – in fact, for a general point, the scheme is just a union of r distinct points. A modern way to phrase it is: the smoothable rank is equal to the border rank for most polynomials. This property is very useful for studying secant varieties, especially, whenever the smoothable rank is equal to the border rank for all points of the secant variety in question. In this note, we investigate those special points for which the smoothable rank is not equal to the border rank. In particular, we show an explicit example of a cubic in five variables with border rank 5 and smoothable rank 6. We also prove that all cubics in at most four variables have the smoothable rank equal to the border rank.
We exhibit a numerical method to compute three-point branched covers of the complex projective line. We develop algorithms for working explicitly with Fuchsian triangle groups and their finite-index subgroups, and we use these algorithms to compute power series expansions of modular forms on these groups.
We study new families of curves that are suitable for efficiently parametrizing their moduli spaces. We explicitly construct such families for smooth plane quartics in order to determine unique representatives for the isomorphism classes of smooth plane quartics over finite fields. In this way, we can visualize the distributions of their traces of Frobenius. This leads to new observations on fluctuations with respect to the limiting symmetry imposed by the theory of Katz and Sarnak.
We give an equivalent definition of the local volume of an isolated singularity
$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\rm Vol}_{\text {BdFF}}(X,0)$
given in [S. Boucksom, T. de Fernex, C. Favre, The volume of an isolated singularity. Duke Math. J. 161 (2012), 1455–1520] in the
$\mathbb{Q}$
-Gorenstein case and we generalize it to the non-
$\mathbb{Q}$
-Gorenstein case. We prove that there is a positive lower bound depending only on the dimension for the non-zero local volume of an isolated singularity if
$X$
is Gorenstein. We also give a non-
$\mathbb{Q}$
-Gorenstein example with
${\rm Vol}_{\text {BdFF}}(X,0)=0$
, which does not allow a boundary
$\Delta $
such that the pair
$(X,\Delta )$
is log canonical.
We present an improved algorithm for the computation of Zariski chambers on algebraic surfaces. The new algorithm significantly outperforms the currently available method and therefore allows us to treat surfaces of high Picard number, where huge numbers of chambers occur. As an application, we efficiently compute the number of chambers supported by the lines on the Segre–Schur quartic.