Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ be a complex smooth projective surface and $L$ be a line bundle on $S$. For any given collection of isolated topological or analytic singularity types, we show the number of curves in the linear system $|L|$ with prescribed singularities is a universal polynomial of Chern numbers of $L$ and $S$, assuming $L$ is sufficiently ample. More generally, we show for vector bundles of any rank and smooth varieties of any dimension, similar universal polynomials also exist and equal the number of singular subvarieties cutting out by sections of the vector bundle. This work is a generalization of Göttsche’s conjecture.

A Brahmagupta quadrilateral is a cyclic quadrilateral whose sides, diagonals and area are all integer values. In this article, we characterise the notions of Brahmagupta, introduced by K. R. S. Sastry [‘Brahmagupta quadrilaterals’, Forum Geom. 2 (2002), 167–173], by means of elliptic curves. Motivated by these characterisations, we use Brahmagupta quadrilaterals to construct infinite families of elliptic curves with torsion group $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ having ranks (at least) four, five and six. Furthermore, by specialising we give examples from these families of specific curves with rank nine.

Let $Q(N;q,a)$ be the number of squares in the arithmetic progression $qn+a$, for $n=0$,$1,\ldots,N-1$, and let $Q(N)$ be the maximum of $Q(N;q,a)$ over all non-trivial arithmetic progressions $qn + a$. Rudin’s conjecture claims that $Q(N)=O(\sqrt{N})$, and in its stronger form that $Q(N)=Q(N;24,1)$ if $N\ge 6$. We prove the conjecture above for $6\le N\le 52$. We even prove that the arithmetic progression $24n+1$ is the only one, up to equivalence, that contains $Q(N)$ squares for the values of $N$ such that $Q(N)$ increases, for $7\le N\le 52$ ($N=8,13,16,23,27,36,41$ and $52$).

Supplementary materials are available with this article.

We consider the natural $A_{\infty }$-structure on the $\mathrm{Ext}$-algebra $\mathrm{Ext}^*(G,G)$ associated with the coherent sheaf $G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$ on a smooth projective curve $C$, where $p_1,\ldots,p_n\in C$ are distinct points. We study the homotopy class of the product $m_3$. Assuming that $h^0(p_1+\cdots +p_n)=1$, we prove that $m_3$ is homotopic to zero if and only if $C$ is hyperelliptic and the points $p_i$ are Weierstrass points. In the latter case we show that $m_4$ is not homotopic to zero, provided the genus of $C$ is greater than $1$. In the case $n=g$ we prove that the $A_{\infty }$-structure is determined uniquely (up to homotopy) by the products $m_i$ with $i\le 6$. Also, in this case we study the rational map $\mathcal{M}_{g,g}\to \mathbb{A}^{g^2-2g}$ associated with the homotopy class of $m_3$. We prove that for $g\ge 6$ it is birational onto its image, while for $g\le 5$ it is dominant. We also give an interpretation of this map in terms of tangents to $C$ in the canonical embedding and in the projective embedding given by the linear series $|2(p_1+\cdots +p_g)|$.

We show that there is no stable Siegel modular form that vanishes on every moduli space of curves.

We prove that if $y''=f(y,y',t,\alpha ,\beta ,\ldots)$ is a generic Painlevé equation from among the classes II, IV and V, and if $y_1,\ldots,y_n$ are distinct solutions, then $\mathrm{tr.deg}(\mathbb{C}(t)(y_1,y'_1,\ldots,y_n,y'_n)/\mathbb{C}(t))=2n$. (This was proved by Nishioka for the single equation $P_{{\rm I}}$.) For generic Painlevé III and VI, we have a slightly weaker result: $\omega $-categoricity (in the sense of model theory) of the solution space, as described below. The results confirm old beliefs about the Painlevé transcendents.

We give a new, geometric proof of the section conjecture for fixed points of finite group actions on projective curves of positive genus defined over the field of complex numbers, as well as its natural nilpotent analogue. As a part of our investigations we give an explicit description of the abelianised section map for groups of prime order in this setting. We also show a version of the $2$-nilpotent section conjecture.

In this work, we describe a method to construct the generic braid monodromy of the preimage of a curve by a Kummer cover. This method is interesting since it combines two techniques, namely, the construction of a highly non-generic braid monodromy and a systematic method to go from a non-generic to a generic braid monodromy. The latter process, called generification, is independent from Kummer covers, and it can be applied in more general circumstances since non-generic braid monodromies appear more naturally and are oftentimes much easier to compute. Explicit examples are computed using these techniques.

We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.

The purpose of this paper is twofold. We present first a vanishing theorem for families of linear series with base ideal being a fat points ideal. We then apply this result in order to give a partial proof of a conjecture raised by Bocci, Harbourne and Huneke concerning containment relations between ordinary and symbolic powers of planar point ideals.

We investigate parabolic Higgs bundles and $\Gamma $-Higgs bundles on a smooth complex projective variety.

Let us consider the locus in the moduli space of curves of genus $2k$ defined by curves with a pencil of degree $k$. Since the Brill–Noether number is equal to $- 2$, such a locus has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves.

This paper studies ‘pro-excision’ for the $K$-theory of one-dimensional, usually semi-local, rings and its various applications. In particular, we prove Geller’s conjecture for equal characteristic rings over a perfect field of finite characteristic, give results towards Geller’s conjecture in mixed characteristic, and we establish various finiteness results for the $K$-groups of singularities, covering both orders in number fields and singular curves over finite fields.

Let $k$ be a locally compact complete field with respect to a discrete valuation $v$. Let $ \mathcal{O} $ be the valuation ring, $\mathfrak{m}$ the maximal ideal and $F(x)\in \mathcal{O} [x] $ a monic separable polynomial of degree $n$. Let $\delta = v(\mathrm{Disc} (F))$. The Montes algorithm computes an OM factorization of $F$. The single-factor lifting algorithm derives from this data a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$, for a prescribed precision $\nu $. In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of $O({n}^{2+ \epsilon } + {n}^{1+ \epsilon } {\delta }^{2+ \epsilon } + {n}^{2} {\nu }^{1+ \epsilon } )$ word operations for the complexity of the computation of a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$, assuming that the residue field of $k$ is small.

We prove that the quotient by ${\mathrm{SL} }_{2} \times {\mathrm{SL} }_{2} $ of the space of bidegree $(a, b)$ curves on ${ \mathbb{P} }^{1} \times { \mathbb{P} }^{1} $ is rational when $ab$ is even and $a\not = b$.

We study natural families of $\bar {\partial } $-operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.

In the recent paper by Pakovich and Muzychuk [Solution of the polynomial moment problem, Proc. Lond. Math. Soc. (3) 99 (2009), 633–657] it was shown that any solution of ‘the polynomial moment problem’, which asks to describe polynomials $Q$ orthogonal to all powers of a given polynomial $P$ on a segment, may be obtained as a sum of so-called ‘reducible’ solutions related to different decompositions of $P$ into a composition of two polynomials of lower degrees. However, the methods of that paper do not permit us to estimate the number of necessary reducible solutions or to describe them explicitly. In this paper we provide a description of polynomial solutions of the functional equation $P_1\circ W_1=P_2\circ W_2=\cdots =P_r\circ W_r,$and on this base describe solutions of the polynomial moment problem in an explicit form suitable for applications.

We study the distribution of the size of Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with a non-trivial $2$-torsion point over $\mathbb {Q}$. This complements the work [Xiong and Zaharescu, Distribution of Selmer groups of quadratic twists of a family of elliptic curves. Adv. Math.219 (2008), 523–553] which studied the same subject for elliptic curves with full 2-torsions over $\mathbb {Q}$ and generalizes [Feng and Xiong, On Selmer groups and Tate–Shafarevich groups for elliptic curves $y^2=x^3-n^3$. Mathematika 58 (2012), 236–274.] for the special elliptic curves $y^2=x^3-n^3$. It is shown that the 2-ranks of these groups all follow the same distribution and in particular, the mean value is $\sqrt {\frac {1}{2}\log \log X}$ for square-free positive integers $n \le X$ as $X \to \infty $.

Given an elliptic curve E over a field of positive characteristic p, we consider how to efficiently determine whether E is ordinary or supersingular. We analyze the complexity of several existing algorithms and then present a new approach that exploits structural differences between ordinary and supersingular isogeny graphs. This yields a simple algorithm that, given E and a suitable non-residue in 𝔽p2, determines the supersingularity of E in O(n3log 2n) time and O(n) space, where n=O(log p) . Both these complexity bounds are significant improvements over existing methods, as we demonstrate with some practical computations.

We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic T7, T8 singularities. We define discrete symplectic invariants (the Lagrangian tangency orders) and compare them with the index of isotropy. We use these invariants to distinguish symplectic singularities of classical T7 singularity. We also give the geometric description of symplectic classes of the singularity.