About

Dragos Oprea received his Ph.D. in Mathematics from the Massachusetts Institute of Technology in 2005. He was a Samuelson Fellow and Szego Assistant Professor at Stanford University from 2005 to 2008. Oprea's research is in algebraic geometry, with a focus on moduli theory and enumerative geometry.

Research topics

• Pure mathematics
• Geometry
• Mathematical analysis
• Mathematics

Selected publications

• On the cohomology of tautological bundles over Quot schemes of curves

  Algebra & Number Theory · 2026-05-06

  preprintOpen access

  We consider tautological bundles and their exterior and symmetric powers on the Quot scheme over the projective line.We prove and conjecture several statements regarding the vanishing of their higher cohomology, and we describe their spaces of global sections via tautological constructions.To this end, we make use of the embedding of the Quot scheme as an explicit local complete intersection in the product of two Grassmannians, studied by Strmme.This allows us to construct resolutions with vanishing cohomology for the tautological bundles and their exterior and symmetric powers.We further illustrate our approach with a few additional cohomological calculations.

  PublisherOA PDFDOI

• Tautological and non-tautological cycles on the moduli space of Abelian varieties

  Inventiones mathematicae · 2025-09-09 · 2 citations

  articleOpen access

  Abstract The tautological Chow ring of the moduli space $\mathcal{A}_{g}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> of principally polarized abelian varieties of dimension $g$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from $\mathcal{A}_{g}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>g</mml:mi> </mml:msub> </mml:math> to the moduli space $\mathcal{M}_{g}^{\operatorname{ct}}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>g</mml:mi> <mml:mo>ct</mml:mo> </mml:msubsup> </mml:math> of genus $g$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> of curves of compact type, we prove that the product class $[\mathcal{A}_{1}\times \mathcal{A}_{5}]\in \mathsf{CH}^{5}( \mathcal{A}_{6})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:mo>]</mml:mo> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>CH</mml:mi> <mml:mn>5</mml:mn> </mml:msup> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>6</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:math> is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the tautological ring $\mathsf{R}^{*}(\mathcal{M}_{6}^{\operatorname{ct}})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mn>6</mml:mn> <mml:mo>ct</mml:mo> </mml:msubsup> <mml:mo>)</mml:mo> </mml:math> in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring $\mathsf{R}^{*}(\mathcal{M}_{6}^{\operatorname{ct}})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mn>6</mml:mn> <mml:mo>ct</mml:mo> </mml:msubsup> <mml:mo>)</mml:mo> </mml:math> has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of $[\mathcal{A}_{1}\times \mathcal{A}_{5}]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>5</mml:mn> </mml:msub> <mml:mo>]</mml:mo> </mml:math> . More generally, the Torelli pullback of the difference between $[\mathcal{A}_{1}\times \mathcal{A}_{g-1}]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>[</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>]</mml:mo> </mml:math> and its tautological projection always lies in the Gorenstein kernel of $\mathsf{R}^{*}(\mathcal{M}_{g}^{\operatorname{ct}})$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>(</mml:mo> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>g</mml:mi> <mml:mo>ct</mml:mo> </mml:msubsup> <mml:mo>)</mml:mo> </mml:math> . The product map $\mathcal{A}_{1}\times \mathcal{A}_{g-1}\rightarrow \mathcal{A}_{g}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>1</mml:mn> </mml:msub>

  PublisherOA PDFDOI

• Tautological projection for cycles on the moduli space of abelian varieties

  Algebraic geometry · 2025-10-24 · 1 citations

  articleOpen access
  PublisherDOI

• Tautological projection for cycles on the moduli space of abelian varieties

  arXiv (Cornell University) · 2024-01-28

  preprintOpen access

  We define a tautological projection operator for algebraic cycle classes on the moduli space of principally polarized abelian varieties $\mathcal{A}_g$: every cycle class decomposes canonically as a sum of a tautological and a non-tautological part. The main new result required for the definition of the projection operator is the vanishing of the top Chern class of the Hodge bundle over the boundary $\bar{\mathcal{A}}_g\smallsetminus \mathcal{A}_g$ of any toroidal compactification $\bar{\mathcal{A}}_g$ of the moduli space $\mathcal{A}_g$. We prove the vanishing by a careful study of residues in the boundary geometry. The existence of the projection operator raises many natural questions about cycles on $\mathcal{A}_g$. We calculate the projections of all product cycles $\mathcal{A}_{g_1}\times \ldots \times \mathcal{A}_{g_\ell}$ in terms of Schur determinants, discuss Faber's earlier calculations related to the Torelli locus, and state several open questions. The Appendix contains a conjecture about the projection of the locus of abelian varieties with real multiplication.

  PublisherOA PDFDOI

• Tautological and non-tautological cycles on the moduli space of abelian varieties

  arXiv (Cornell University) · 2024-08-16

  preprintOpen access

  The tautological Chow ring of the moduli space $\mathcal{A}_g$ of principally polarized abelian varieties of dimension $g$ was defined and calculated by van der Geer in 1999. By studying the Torelli pullback of algebraic cycles classes from $\mathcal{A}_g$ to the moduli space $\mathcal{M}_g^{\mathrm{ct}}$ of genus $g$ of curves of compact type, we prove that the product class $[\mathcal{A}_1\times \mathcal{A}_5]\in \mathsf{CH}^{5}(\mathcal{A}_6)$ is non-tautological, the first construction of an interesting non-tautological algebraic class on the moduli spaces of abelian varieties. For our proof, we use the complete description of the the tautological ring $\mathsf{R}^*(\mathcal{M}_6^{\mathrm{ct}})$ in genus 6 conjectured by Pixton and recently proven by Canning-Larson-Schmitt. The tautological ring $\mathsf{R}^*(\mathcal{M}_6^{\mathrm{ct}})$ has a 1-dimensional Gorenstein kernel, which is geometrically explained by the Torelli pullback of $[\mathcal{A}_1\times \mathcal{A}_5]$. More generally, the Torelli pullback of the difference between $[\mathcal{A}_1\times \mathcal{A}_{g-1}]$ and its tautological projection always lies in the Gorenstein kernel of $\mathsf{R}^*(\mathcal{M}_g^{\mathrm{ct}})$. The product map $\mathcal{A}_1\times \mathcal{A}_{g-1}\rightarrow \mathcal{A}_g$ is a Noether-Lefschetz locus with general Neron-Severi rank 2. A natural extension of van der Geer's tautological ring is obtained by including more general Noether-Lefschetz loci. Results and conjectures related to cycle classes of Noether-Lefschetz loci for all $g$ are presented.

  PublisherOA PDFDOI

• The Chow ring of the moduli space of degree 2 quasi-polarized K3 surfaces

  Journal für die reine und angewandte Mathematik (Crelles Journal) · 2024-08-27

  article

  Abstract We study the Chow ring with rational coefficients of the moduli space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \mathcal{F}_{2} of quasi-polarized K3 surfaces of degree 2. We find generators, relations, and calculate the Chow Betti numbers. The highest nonvanishing Chow group is <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mi mathvariant="sans-serif">A</m:mi> <m:mn>17</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>≅</m:mo> <m:mi mathvariant="double-struck">Q</m:mi> </m:mrow> </m:math> \mathsf{A}^{17}(\mathcal{F}_{2})\cong{\mathbb{Q}} . We prove that the Chow ring consists of tautological classes and is isomorphic to the even cohomology. The Chow ring is not generated by divisors and does not satisfy duality with respect to the pairing into <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="sans-serif">A</m:mi> <m:mn>17</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathsf{A}^{17}(\mathcal{F}_{2}) . The kernel of the pairing is a 1-dimensional subspace of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi mathvariant="sans-serif">A</m:mi> <m:mn>9</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \mathsf{A}^{9}(\mathcal{F}_{2}) which we calculate explicitly. In the appendix, we revisit Kirwan–Lee’s calculation of the Poincaré polynomial of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">F</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \mathcal{F}_{2} .

  PublisherDOI

• Euler characteristics of tautological bundles over Quot schemes of curves

  Advances in Mathematics · 2023-03-10 · 3 citations

  article1st authorCorresponding
  PublisherDOI

• The Chow ring of the moduli space of degree $2$ quasi-polarized K3 surfaces

  arXiv (Cornell University) · 2023-07-18

  preprintOpen access

  We study the Chow ring with rational coefficients of the moduli space $\mathcal F_{2}$ of quasi-polarized $K3$ surfaces of degree $2$. We find generators, relations, and calculate the Chow Betti numbers. The highest nonvanishing Chow group is $\mathsf A^{17}(\mathcal F_2)\cong {\mathbb{Q}}$. We prove that the Chow ring consists of tautological classes and is isomorphic to the even cohomology. The Chow ring is not generated by divisors and does not satisfy duality with respect to the pairing into $\mathsf A^{17}(\mathcal F_2)$. The kernel of the pairing is a 1-dimensional subspace of $\mathsf{A}^{9}(\mathcal F_2)$ which we calculate explicitly. In the appendix, we revisit Kirwan-Lee's calculation of the Poincaré polynomial of $\mathcal{F}_2$.

  PublisherOA PDFDOI

• Big and Nef Tautological Vector Bundles over the Hilbert Scheme of Points

  Symmetry Integrability and Geometry Methods and Applications · 2022-08-12 · 1 citations

  articleOpen access1st authorCorresponding

  We study tautological vector bundles over the Hilbert scheme of points on surfaces. For each K-trivial surface, we write down a simple criterion ensuring that the tautological bundles are big and nef, and illustrate it by examples. In the K3 case, we extend recent constructions and results of Bini, Boissire and Flamini from the Hilbert scheme of 2 and 3 points to an arbitrary number of points. Among the K-trivial surfaces, the case of Enriques surfaces is the most involved. Our techniques apply to other smooth projective surfaces, including blowups of K3s and minimal surfaces of general type, as well as to the punctual Quot schemes of curves.

  PublisherOA PDFDOI

• Euler characteristics of tautological bundles over Quot schemes of curves

  arXiv (Cornell University) · 2022-07-04

  preprintOpen access1st authorCorresponding

  We compute the Euler characteristics of tautological vector bundles and their exterior powers over the Quot schemes of curves. We give closed-form expressions over punctual Quot schemes in all genera. For higher rank quotients of a trivial vector bundle, we obtain answers in genus zero. We also study the Euler characteristics of the symmetric powers of the tautological bundles, for rank zero quotients.

  PublisherOA PDFDOI

Recent grants

• CAREER: Stable sheaves, stable quotients, stable pairs

  NSF · $445k · 2012–2019

• Moduli spaces of morphisms and sheaves

  NSF · $74k · 2008–2010

• The geometry of the moduli spaces of morphisms and sheaves

  NSF · $150k · 2010–2015

• Moduli Spaces, Tautological Rings, and Theta Functions

  NSF · $265k · 2018–2023

Frequent coauthors

• Alina Marian

  44 shared

• Rahul Pandharipande

  28 shared

• Samir Canning

  ETH Zurich

  4 shared

• Dimitri Zvonkine

  Université de Versailles Saint-Quentin-en-Yvelines

  4 shared

• Barbara Bolognese

  Roma Tre University

  4 shared

• Aaron Pixton

  University of Michigan–Ann Arbor

  3 shared

• Drew Johnson

  Brigham Young University

  3 shared

• Kōta Yoshioka

  Kobe University

  2 shared

Awards & honors

• Hellman Fellowship (2010)
• Alfred P. Sloan Research Fellowship (2011)
• NSF CAREER Grant (2012)