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Register a new userDynamical invariants and intersection theory on the flex and gothic loci
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Authors
Dawei Chen
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The flex locus parameterizes plane cubics with three collinear cocritical points under a projection, and the gothic locus arises from quadratic differentials with zeros at a fiber of the projection and with poles at the cocritical points. The flex and gothic loci provide the first example of a primitive, totally geodesic subvariety of moduli space and new ${rm SL}_2(mathbb{R})$-invariant varieties in Teichmuller dynamics, as discovered by McMullen-Mukamel-Wright. In this paper we determine the divisor class of the flex locus as well as various tautological intersection numbers on the gothic locus. For the case of the gothic locus our result confirms numerically a conjecture of Chen-Moller-Sauvaget about computing sums of Lyapunov exponents for ${rm SL}_2(mathbb{R})$-invariant varieties via intersection theory.
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We explain a general construction of double covers of quadratic degeneracy loci and Lagrangian intersection loci based on reflexive sheaves. We relate the double covers of quadratic degeneracy loci to the Stein factorizations of the relative Hilbert schemes of linear spaces of the corresponding quadric fibrations. We give a criterion for these double covers to be nonsingular. As applications of these results, we show that the double covers of the EPW sextics obtained by our construction give OGradys double EPW sextics and that an analogous construction gives Iliev-Kapustka-Kapustka-Ranestads EPW cubes.
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An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of algebraic geometry, dynamical systems, geometry and topology lead to the study of such objects. It comes as a natural consequence that we can employ in our study algebraic, analytic, combinatorial and dynamical perspectives. These lecture notes aim to provide an expository introduction to this subject that will emphasize the aforementioned links between different areas of mathematics. We will associate to an Abelian differential a flat surface with conical singularities such that the underlying Riemann surface is obtained from a polygon by identifying edges with one another via translation. We will focus on studying these objects in families and describe some properties of the orbit as we vary the polygon by the action of $GL_2^{+}(mathbb{R})$ on the plane.
We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let $N geq 2$, and consider an isolated complete intersection curve singularity germ $f colon (mathbb{C}^N,0) to (mathbb{C}^{N-1},0)$. We introduce a numerical function $m mapsto operatorname{AD}_{(2)}^m(f)$ that arises as an error term when counting $m^{mathrm{th}}$-order weight-$2$ inflection points with ramification sequence $(0, dots, 0, 2)$ in a $1$-parameter family of curves acquiring the singularity $f = 0$, and we compute $operatorname{AD}_{(2)}^m(f)$ for various $(f,m)$. Particularly, for a node defined by $f colon (x,y) mapsto xy$, we prove that $operatorname{AD}_{(2)}^m(xy) = {{m+1} choose 4},$ and we deduce as a corollary that $operatorname{AD}_{(2)}^m(f) geq (operatorname{mult}_0 Delta_f) cdot {{m+1} choose 4}$ for any $f$, where $operatorname{mult}_0 Delta_f$ is the multiplicity of the discriminant $Delta_f$ at the origin in the deformation space. Furthermore, we show that the function $m mapsto operatorname{AD}_{(2)}^m(f) -(operatorname{mult}_0 Delta_f) cdot {{m+1} choose 4}$ is an analytic invariant measuring how much the singularity counts as an inflection point. We obtain similar results for weight-$2$ inflection points with ramification sequence $(0, dots, 0, 1,1)$ and for weight-$1$ inflection points, and we apply our results to solve various related enumerative problems.
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