In this paper, we consider the following question: how many degree $d$ curves are there in $\mathbb{P}^3$ (passing through the right number of generic lines and points), whose image lies inside a $\mathbb{P}^2$, having $δ$ nodes and one singularity of codimension $k$. We obtain an explicit formula for this number when $δ+k \leq 4$ (i.e. the total codimension of the singularities is not more than four). We use a topological method to compute the degenerate contribution to the Euler class; it is an extension of the method that originates in a paper by A. Zinger and which is further pursued by S. Basu and the second author. Using this method, we have obtained formulas when the singularities present are more degenerate than nodes (such as cusps, tacnodes and triple points). When the singularities are only nodes, we have verified that our answers are consistent with those obtained by by S. Kleiman and R. Piene and by T. Laarakker. We also verify that our answer for the characteristic number of planar cubics with a cusp and the number of planar quartics with two nodes and one cusp is consistent with the answer obtained by R. Singh and the second author, where they compute the characteristic number of rational planar curves in $\mathbb{P}^3$ with a cusp. We also verify some of the numbers predicted by the conjecture made by Pandharipande, regarding the enumerativity of BPS numbers for $\mathbb{P}^3$.
We prove the constancy of the Łojasiewicz exponent in non-degenerate $μ$-constant deformations of surface singularities. This is a positive answer to a question posed by B. Teissier.
This paper has been withdrawn. Consider an isolated complex hypersurface singularity, f(x_1,..,x_n)=0. For Newton-non-degenerate singularities the local topology is completely determined by an associated polyhedral object, the Newton diagram. "Most" singularities are not Newton-non-degenerate, for any choice of local coordinates. An old question of Arnol'd asks whether for any hypersurface singularity there exists a stabilization, f(x_1,...,x_n)+z^2_1+...+z^2_r, that becomes Newton-non-degenerate after some change of coordinates. The answer is: "totally no". We give some simple obstructions an…
The jump of the Milnor number of an isolated singularity $f_0$ is the minimal non-zero difference between the Milnor numbers of $f_0$ and one of its deformations $(f_s)$. We give a formula for the jump in some class of surface singularities in the case deformations are non-degenerate.
We give a formula for the spectral pairs (after Steenbrink) for composite singularities of several variables. (Note that for two variable case is studyed by Nemethi-Steenbrink.) Here composite singularity is given by the equation f(g_1, ..., g_n) = 0. For technical reason, we assume that f is non-degenerated with respect to the Newton boundary.
We study the Weierstrass preparation and division theorems over arbitrary test rings and the local structure of singularities in the space of nondegenerate arcs on algebraic varieties. As an application, we prove a strengthened version of a theorem of Grinberg-Kazhdan and Drinfeld.
We give a criterion to test geometric properties such as Whitney equisingularity and Thom's $a_f$ condition for new families of (possibly non-isolated) hypersurface singularities that "behave well" with respect to their Newton diagrams. As an important corollary, we obtain that in such families all members have isomorphic Milnor fibrations.
The jump of the Milnor number of an isolated singularity $f_{0}$ is the minimal non-zero difference between the Milnor numbers of $f_{0}$ and one of its deformation $(f_{s}).$ In the case $f_{s}$ are non-degenerate singularities we call the jump non-degenerate. We give a formula (an inductive algorithm using diophantine equations) for the non-degenerate jump of $f_{0}$ in the case $f_{0}$ is a convenient singularity with only one $(n-1)$-dimensional face of its Newton diagram which equivalently (in our problem) can be replaced by the Brieskorn-Pham singularities.
The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations -- the Denef--Loeser conjecture on topological zeta functions -- is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables. A crucial difference from the known case of thr…
We give formulas for the degrees of the spaces of foliations in P2 with a dicritical singularity of prescribed order. Blowing up such singularity induces, generically, a foliation with all but finitely many leaves transversal to the exceptional line; we also find the degree of the locus defined by imposing a leaf of total contact with the exceptional line.
Let $f$ be an isolated singularity at the origin of $\mathbb{C}^n$. One of many invariants that can be associated with $f$ is its Łojasiewicz exponent $\mathcal{L}_0 (f)$, which measures, to some extent, the topology of $f$. We give, for generic surface singularities $f$, an effective formula for $\mathcal{L}_0 (f)$ in terms of the Newton polyhedron of $f$. This is a realization of one of Arnold's postulates.
We recover the Newton diagram (modulo a natural ambiguity) from the link for any surface hypersurface singularity with non-degenerate Newton principal part whose link is a rational homology sphere. As a corollary, we show that the link determines the embedded topological type, the Milnor fibration, and the multiplicity of such a germ. This proves (even a stronger version of) Zariski's Conjecture about the multiplicity for such a singularity.
In 2002, Alexeev provided a modular interpretation for the toroidal compactification of $A_g$ for the 2nd Voronoi fan. In this paper we show that pairs $(X,Θ)$ in the boundary of this moduli space are semi-log canonical, the analog of log canonical in the non-normal setting. This extends the result of Koll${á}$r stating that principally polarized abelian pairs $(X,Θ)$ are log canonical.
We construct and study the reduced, relative, genus one Gromov--Witten theory of very ample pairs. These invariants form the principal component contribution to relative Gromov--Witten theory in genus one and are relative versions of Zinger's reduced Gromov--Witten invariants. We relate the relative and absolute theories by degeneration of the tangency conditions, and the resulting formulas generalise a well-known recursive calculation scheme put forward by Gathmann in genus zero. The geometric input is a desingularisation of the principal component of the moduli space of genus one logarithmic…
The Milnor number, μ(X,0), and the singularity genus, p_g(X,0), are fundamental invariants of isolated hypersurface singularities (more generally, of local complete intersections). The long standing Durfee conjecture (and its generalization) predicted the inequality μ(X,0) \geq (n+1)!p_g(X,0), here n=dim(X,0). Recently we have constructed counterexamples, proposed a corrected bound and verified it for the homogeneous complete intersections. In the current paper we treat the case of germs with Newton-non-degenerate principal part when the Newton diagrams are "large enough", i.e. they are large …
In his groundbreaking work on classification of singularities with regard to right and stable equivalence of germs, Arnold has listed normal forms for all isolated hypersurface singularities over the complex numbers with either modality less than or equal to two or Milnor number less than or equal to 16. Moreover, he has described an algorithmic classifier, which determines the type of a given such singularity. In the present paper, we extend Arnold's work to a large class of singularities which is unbounded with regard to modality and Milnor number. We develop an algorithmic classifier, which…
We introduce two generalizations of Newton-non-degenerate (Nnd) singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called topologically Newton-non-degenerate (tNnd) if the local embedded topological singularity type can be restored from a collection of Newton diagrams (for some coordinate choices). A singularity that is not tNnd is called essentially Newton-degenerate. For plane curves we give an explicit characterization of tNnd singularities; for hypersurfaces we provide several examples. Next, we treat the question: whether Nnd or tNnd is a property of…
We establish Kyoji Saito's continuous limit distribution for the spectrum of Newton non-degenerate hypersurface singularities. Investigating Saito's notion of dominant value in the case of irreducible plane curve singularities, we find that the log canonical threshold is strictly bounded below by the doubled inverse of the Milnor number. We show that this bound is asymptotically sharp.
We study the connection between the singularities of a finite type $\mathbb{Z}$-scheme X and the asymptotic point count of X over various finite rings. In particular, if the generic fiber $X_{\mathbb{Q}}=X\times_{\mathrm{Spec}\mathbb{Z}}\mathrm{Spec}\mathbb{Q}$ is a local complete intersection, we show that the boundedness of $\frac{\left|X(\mathbb{Z}/p^{n}\mathbb{Z})\right|}{p^{n\mathrm{dim}X_{\mathbb{Q}}}}$ in p and n is in fact equivalent to the condition that $X_{\mathbb{Q}}$ is reduced and has rational singularities. This paper completes a result of Aizenbud and Avni.
We prove some combinatorial results required for the proof of the following conjecture of Nekrasov: The generating function of closed string invariants in local Calabi-Yau geometries obtained by appropriate fibrations of $A_N$ singularities over $P^1$ reproduce the generating function of equivariant $\hat{A}$-genera of moduli space of instants on $C^2$.
We study the topology of the complements of discriminants of simple real boundary singularities by counting the connected components of these sets and assigning to them certain topological characteristics. Results of this paper serve as a generalization of those recently acquired by Vassiliev arXiv:2109.12287 for ordinary function singularities.
Let $Γ$ be a finite subgroup of $\mathrm{Sp}(V)$. In this article we count the number of symplectic resolutions admitted by the quotient singularity $V / Γ$. Our approach is to compare the universal Poisson deformation of the symplectic quotient singularity with the deformation given by the Calogero-Moser space. In this way, we give a simple formula for the number of $\mathbb{Q}$-factorial terminalizations admitted by the symplectic quotient singularity in terms of the dimension of a certain Orlik-Solomon algebra naturally associated to the Calogero-Moser deformation. This dimension is explici…
Let f_0 be a plane curve singularity. We study the Minor numbers of singularities in deformations of f_0. We completely describe the set of these Milnor numbers for homogeneous singularities f_0 in the case of non-degenerate deformations and obtain some partial results on this set in the general case.
We prove that for two germs of analytic mappings $f,g\colon (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^p,0)$ with the same Newton polyhedra which are (Khovanskii) non-degenerate and their zero sets are complete intersections with isolated singularity at the origin, there is a piecewise analytic family $\{f_t\}$ of analytic maps with $f_0=f, f_1=g$ which has a so-called {\it uniform stable radius for the Milnor fibration}. As a corollary, we show that their Milnor numbers are equal. Also, a formula for the Milnor number is given in terms of the Newton polyhedra of the component functions. This is…
Let $X\to \mathbb P^2$ be the elliptic Calabi-Yau threefold given by a general Weierstrass equation. We answer the enumerative question of how many discrete rational curves lie over lines in the base, proving part of a conjecture by Huang, Katz, and Klemm. The key inputs are a modularity theorem of Kudla and Millson for locally symmetric spaces of orthogonal type and the deformation theory of $A_n$ singularities.
We obtain a recursive formula for the characteristic number of degree $d$ curves in $\mathbb{P}^2$ with prescribed singularities (of type $A_k$) that are tangent to a given line. The formula is in terms of the characteristic number of curves with exactly those singularities. Combined with the results of S. Basu and R. Mukherjee, this gives us a complete formula for the characteristic number of curves with $δ$-nodes and one singularity of type $A_k$, tangent to a given line, provided $δ+ k \leq 8$. We use a topological method, namely the method of dynamic intersections to compute the degenerate…
This paper studies hypersurface exceptional singularities in $\mathbb C^n$ defined by non-degenerate function. For each canonical hypersurface singularity, there exists a weighted homogeneous singularity such that the former is exceptional if and only if the latter is exceptional. So we study the weighted homogeneous case and prove that the number of weights of weighted homogeneous exceptional singularities are finite. Then we determine all exceptional singularities of the Brieskorn type of dimension 3.
We study the counting function of topological Poincaré series associated with rational homology sphere plumbed 3-manifold with connected negative definite tree, interpreting as an alternating sum of coefficient functions associated with some Taylor expansions. It is motivated by a theorem of Szenes and Vergne which expresses these coefficient functions in terms of Jeffrey--Kirwan residues. This is used to prove the uniqueness of the quasipolynomiality inside a special cone, the structure of the counting function in terms of the graph and construction for a polynomial generalization of the Seib…
We study the curve counting invariants of Calabi--Yau 3-folds via the Weyl reflection along a ruled divisor. We obtain a new rationality result and functional equation for the generating functions of Pandharipande--Thomas invariants. When the divisor arises as resolution of a curve of $A_1$-singularities, our results match the rationality of the associated Calabi--Yau orbifold. The symmetry on generating functions descends from the action of an infinite dihedral group of derived auto-equivalences, which is generated by the derived dual and a spherical twist. Our techniques involve wall-crossin…
For a convenient and Newton non-degenerate singularity, the Milnor number is computed from the complement of its Newton diagram in the first quadrant, so-called Kouchnirenko's formula. In this paper, we consider tropical curves dual to subdivisions of this complement for a plane curve singularity and show the existence of a tropical curve satisfying a certain formula, which looks like a well-known formula for a real morsification due to A'Campo and Gusein-Zade.
In a previous paper, there was defined a multi-index filtration on the ring of functions on a hypersurface singularity corresponding to its Newton diagram generalizing (for a curve singularity) the divisorial one. Its Poincaré series was computed for plane curve singularities non-degenerate with respect to their Newton diagrams. Here we use another technique to compute the Poincaré series for plane curve singularities without the assumption that they are non-degenerate with respect to their Newton diagrams. We show that the Poincaré series only depends on the Newton diagram and not on the defi…
Let $X\subseteq \mathbb{P}^N$ be a non-degenerate normal projective variety of codimension $e$ and degree $d$ with isolated $\mathbb{Q}$-Gorenstein singularities. We prove that the Castelnuovo-Mumford regularity $\text{reg}(\mathcal{O}_{X})\le d-e$, as predicted by the Eisenbud-Goto regularity conjecture. Such a bound fails for general projective varieties. The main techniques are Noma's classification of non-degenerated projective varieties and Nadel vanishing for multiplier ideals. We also classify the extremal and the next to extremal cases.
According to the Kouchnirenko theorem, for a generic (precisely non-degenerate in the Kouchnirenko sense) isolated singularity $f$ its Milnor number $μ(f)$ is equal to the Newton number $ν(Γ_{+}(f))$ of a combinatorial object associated to $f$, the Newton polyhedron $Γ_+ (f)$. We give a simple condition characterising, in terms of $Γ_+ (f)$ and $Γ_+ (g)$, the equality $ν(Γ_{+}(f)) = ν(Γ_{+}(g))$, for any surface singularities $f$ and $g$ satisfying $Γ_+ (f) \subset Γ_+ (g)$. This is a complete solution to an Arnold's problem (1982-16) in this case.
We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is arXiv:1509.07453, where a tropical-algebraic correspondence theorem was proved that relates counts of rational curves in toric varieties that satisfy point conditions and cross-ratio constraints to the analogous tropical counts. We proceed in two steps: based on tropical intersection theory we first study tropical cross-ratios and introduce degenerated cross-ratios. Second we provide a lattice path algorithm that produces a…
Burban-Drozd showed that the degenerate cusp singularities have tame Cohen-Macaulay representation type, and classified all indecomposable Cohen-Macaulay modules over them. One of their main example is the non-isolated singularity $W=xyz$. On the other hand, Abouzaid-Auroux-Efimov-Katzarkov-Orlov showed that $W=xyz$ is mirror to a pair of pants. In this paper, we investigate homological mirror symmetry of these indecomposable Cohen-Macaulay modules for $xyz$. Namely, we show that closed geodesics (with a flat $\mathbb{C}$-bundle) of a hyperbolic pair of pants have a one-to-one correspondence w…
We define new partition functions for theories with targets on toric singularities via products of old partition functions on crepant resolutions. We compute explicit examples and show that the new partition functions turn out to be homogeneous on MacMahon factors.
We study the relationship between singularities of finite-dimensional integrable systems and singularities of the corresponding spectral curves. For the large class of integrable systems on matrix polynomials, which is a general framework for various multidimensional spinning tops, as well as Beauville systems, we prove that if the spectral curve is nodal, then all singularities on the corresponding fiber of the system are non-degenerate. We also show that the type of a non-degenerate singularity can be read off from the behavior of double points on the spectral curve under an appropriately de…
In this paper, we determine the bifurcation set of a real polynomial function of two variables for non-degenerate case in the sense of Newton polygons by using a toric compactification. We also count the number of singular phenomena at infinity, called "cleaving" and "vanishing" in the same setting. Finally, we give an upper bound of the number of elements in the bifurcation set in terms of its Newton polygon. To obtain the upper bound, we apply toric modifications to the singularities at infinity successively.
We relate the singularities of a scheme $X$ to the asymptotics of the number of points of $X$ over finite rings. This gives a partial answer to a question of Mustata. We use this result to count representations of arithmetic lattices. More precisely, if $Γ$ is an arithmetic lattice whose $\mathbb{Q}$-rank is greater than one, let $r_n(Γ)$ be the number of irreducible $n$-dimensional representations of $Γ$ up to isomorphism. We prove that there is a constant $C$ (for example, $C=746$ suffices) such that $r_n(Γ)=O(n^C)$ for every such $Γ$. This answers a question of Larsen and Lubotzky.
