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Ravi Vakil

Publications and source records attributed to Ravi Vakil.

At least 19 recordsLinked to original sources
arXiv2026

Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity

We give a quiver description of the space of based algebraic maps from $\mathbb{P}^{1}$ to the Lagrangian Grassmannian (and its orthogonal counterpart). We show our descriptions lead to an algebro-geometric refinement of some of the homotopy equivalences in real Bott periodicity. In particular, we get an isomorphism in Larson and Vakil's ``naive algebro-geometric homotopy category'' whose topological realization (after specializing to $\mathbb{C}$) recovers the classical homotopy equivalences $\Omega^{2}(Sp/U) \simeq BO\times \mathbb{Z}$ and $\Omega^{2}(O/U) \simeq BSp\times \mathbb{Z}$.

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arXiv2026

The motivic class of the space of genus $0$ maps to the flag variety

Let $\operatorname{Fl}_{n+1}$ be the variety of complete flags in $\mathbb{A}^{n+1}$ and let $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ be the space of based maps $f:\mathbb{P}^{1}\to \operatorname{Fl}_{n+1}$ in the class $f_{*}[\mathbb{P}^{1}]=\beta$. We show that under a mild positivity condition on $\beta$, the class of $\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})$ in $K_{0}(\operatorname{Var})$, the Grothendieck group of varieties, is given by \[ [\Omega^{2}_{\beta}(\operatorname{Fl}_{n+1})] = [\operatorname{GL}_{n}\times \mathbb{A}^{a}]. \] The proof of this result was obtained in conjunction with Google Gemini and related tools. We briefly discuss this research interaction, which may be of independent interest. However, the treatment in this paper is entirely human-authored (aside from excerpts in an appendix which are clearly marked as such).

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arXiv2024

Complex Bott Periodicity in algebraic geometry

We state and prove a form of Bott periodicity (for $U(n)$) in an algebraic setting (so, $GL(n)$) which makes sense over $\mathbb{Z}$, which also specializes to Bott periodicity in the usual sense (hence giving yet another proof of classical Bott periodicity). An appendix by B. Church gives a specialization of the constructions and results to motivic homotopy theory, which may be of independent interest.

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arXiv2024

Tschirnhausen bundles of covers of the projective line

A degree $d$ genus $g$ cover of the complex projective line by a smooth curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. Which bundles are possible? Equivalently, which $\mathbb{P}^{d-2}$-bundles over $\mathbb{P}^1$ contain such covers? (In the language of many previous papers: what are the scrollar invariants of the cover?) We give a complete answer in degree $4$, which exhibits the expected pathologies. We describe a polytope (one per degree) which we propose gives the complete answer for primitive covers, i.e. covers that don't factor through a subcover. We show that all such bundles (for primitive covers) lie in this polytope, and that a ``positive proportion'' of the polytope arises from smooth covers. Moreover, we show the necessity of the primitivity assumption. Finally, we show that the image of the map from the Hurwitz space of smooth covers to the space of bundles is not preserved by generization (for $d>5$ and $g \gg_d 1$).

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arXiv2024

The interpolation problem: When can you pass a curve of a given type through N random points in space?

The interpolation problem is a natural and fundamental question whose roots trace back to ancient Greece. The story is long and rich, with many chapters, and a complete solution has been obtained only recently. Exploring it leads us on a tour through a number of general themes in geometry. This concrete problem motivates fundamental concepts such as moduli spaces and their properties, deformation theory, normal bundles, and more. Questions about smooth objects lead us to consider singular (non-smooth) objects, and in fact these smooth objects are studied by instead focusing on somehow simpler "non-smooth" objects, and then deforming them.

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arXiv2022

Low degree Hurwitz stacks in the Grothendieck ring

For $2 \leq d \leq 5$, we show that the class of the Hurwitz space of smooth degree $d$, genus $g$ covers of $\mathbb P^1$ stabilizes in the Grothendieck ring of stacks as $g \to \infty$, and we give a formula for the limit. We also verify this stabilization when one imposes ramification conditions on the covers, and obtain a particularly simple answer for this limit when one restricts to simply branched covers.

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arXiv2021

Numerical Schubert Calculus in Macaulay2

The Macaulay2 package NumericalSchubertCalculus provides methods for the numerical computation of Schubert problems on Grassmannians. It implements both the Pieri homotopy algorithm and the Littlewood-Richardson homotopy algorithm. Each algorithm has two independent implementations in this package. One is in the scripting language of Macaulay2 using the package NumericalAlgebraicGeometry, and the other is in the compiled code of PHCpack.

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arXiv2020

Numerical Schubert Calculus via the Littlewood-Richardson Homotopy Algorithm

We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances with tens of thousands of solutions.

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arXiv2018

Motivic Hilbert zeta functions of curves are rational

The motivic Hilbert zeta function of a variety is the generating function for classes in the Grothendieck ring of varieties of Hilbert schemes of points of the variety. In this paper, the motivic Hilbert zeta function of a reduced curve is shown to be rational.

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arXiv2013

Discriminants in the Grothendieck Ring

We consider the "limiting behavior" of *discriminants*, by which we mean informally the locus in some parameter space of some type of object where the objects have certain singularities. We focus on the space of partially labeled points on a variety X, and linear systems on X. These are connected --- we use the first to understand the second. We describe their classes in the Grothendieck ring of varieties, as the number of points gets large, or as the line bundle gets very positive. They stabilize in an appropriate sense, and their stabilization is given in terms of motivic zeta values. Motivated by our results, we conjecture that the symmetric powers of geometrically irreducible varieties stabilize in the Grothendieck ring (in an appropriate sense). Our results extend parallel results in both arithmetic and topology. We give a number of reasons for considering these questions, and propose a number of new conjectures, both arithmetic and topological.

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arXiv2013

The Chow ring of the moduli space of curves of genus 6

We determine the Chow ring (with Q-coefficients) of M_6 by showing that all Chow classes are tautological. In particular, all algebraic cohomology is tautological, and the natural map from Chow to cohomology is injective. To demonstrate the utility of these methods, we also give quick derivations of the Chow groups of moduli spaces of curves of lower genus. The genus 6 case relies on the particularly beautiful Brill-Noether theory in this case, and in particular on a rank 5 vector bundle "relativizing" a baby case of a celebrated construction of Mukai, which we interpret as a subbundle of the rank 6 vector bundle of quadrics cutting out the canonical curve.

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arXiv2012

Mnev-Sturmfels universality for schemes

We prove a scheme-theoretic version of Mnev-Sturmfels Universality, suitable to be used in the proof of Murphy's Law in Algebraic Geometry. Somewhat more precisely, we show that any singularity type of finite type over Z appears on some incidence scheme of points and lines, subject to some particular further constraints.

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arXiv2011

Formal pseudodifferential operators and Witten's r-spin numbers

We derive an effective recursion for Witten's r-spin intersection numbers, using Witten's conjecture relating r-spin numbers to the Gel'fand-Dikii hierarchy (Theorem 4.1). Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions: Propositions 5.2 and 5.4, Corollary 5.5). We use these closed-form descriptions to prove Harer-Zagier's formula for the Euler characteristic of M_{g,1}. Finally in Section 6, we extend Witten's series expansion formula for the Landau-Ginzburg potential to study r-spin numbers in the small phase space in genus zero. Our key tool is the calculus of formal pseudodifferential operators, and is partially motivated by work of Brezin and Hikami.

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arXiv2011

The geometry of eight points in projective space: Representation theory, Lie theory, dualities

This paper deals with the geometry of the space (GIT quotient) M_8 of 8 points in P^1, and the Gale-quotient N'_8 of the GIT quotient of 8 points in P^3. The space M_8 comes with a natural embedding in P^{13}, or more precisely, the projectivization of the S_8-representation V_{4,4}. There is a single S_8-skew cubic C in P^{13}. The fact that M_8 lies on the skew cubic C is a consequence of Thomae's formula for hyperelliptic curves, but more is true: M_8 is the singular locus of C. These constructions yield the free resolution of M_8, and are used in the determination of the "single" equation cutting out the GIT quotient of n points in P^1 in general. The space N'_8 comes with a natural embedding in P^{13}, or more precisely, PV_{2,2,2,2}. There is a single skew quintic Q containing N'_8, and N'_8 is the singular locus of the skew quintic Q. The skew cubic C and skew quintic Q are projectively dual. (In particular, they are surprisingly singular, in the sense of having a dual of remarkably low degree.) The divisor on the skew cubic blown down by the dual map is the secant variety Sec(M_8), and the contraction Sec(M_8) - - > N'_8 factors through N_8 via the space of 8 points on a quadric surface. We conjecture that the divisor on the skew quintic blown down by the dual map is the quadrisecant variety of N'_8 (the closure of the union of quadrisecant *lines*), and that the quintic Q is the trisecant variety. The resulting picture extends the classical duality in the 6-point case between the Segre cubic threefold and the Igusa quartic threefold. We note that there are a number of geometrically natural varieties that are (related to) the singular loci of remarkably singular cubic hypersurfaces. Some of the content of this paper appeared in arXiv/0809.1233.

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arXiv2011

Universal covering spaces and fundamental groups in algebraic geometry as schemes

In topology, the notions of the fundamental group and the universal cover are closely intertwined. By importing usual notions from topology into the algebraic and arithmetic setting, we construct a fundamental group family from a universal cover, both of which are schemes. A geometric fiber of the fundamental group family (as a topological group) is canonically the 'etale fundamental group. The constructions apply to all connected quasicompact quasiseparated schemes. With different methods and hypotheses, this fundamental group family was already constructed by Deligne.

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arXiv2010

Solving Schubert Problems with Littlewood-Richardson Homotopies

We present a new numerical homotopy continuation algorithm for finding all solutions to Schubert problems on Grassmannians. This Littlewood-Richardson homotopy is based on Vakil's geometric proof of the Littlewood-Richardson rule. Its start solutions are given by linear equations and they are tracked through a sequence of homotopies encoded by certain checker configurations to find the solutions to a given Schubert problem. For generic Schubert problems the number of paths tracked is optimal. The Littlewood-Richardson homotopy algorithm is implemented using the path trackers of the software package PHCpack.

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arXiv2009

The ideal of relations for the ring of invariants of n points on the line

The study of the projective coordinate ring of the (geometric invariant theory) moduli space of n ordered points on P^1 up to automorphisms began with Kempe in 1894, who proved that the ring is generated in degree one in the main (n even, unit weight) case. We describe the relations among the invariants for all possible weights. In the main case, we show that up to the symmetric group symmetry, there is a single equation. For n not 6, it is a simple quadratic binomial relation. (For n=6, it is the classical Segre cubic relation.) For general weights, the ideal of relations is generated by quadratics inherited from the case of 8 points. This paper completes the program set out in [HMSV1].

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