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Bilinear secants and birational geometry of blowups of $\mathbb P^n \times \mathbb P^{n+1}$

This paper introduces bilinear secant varieties to analyze the birational geometry of blowups of $\mathbb{P}^n \times \mathbb{P}^{n+1}$ at general points, proving these varieties are log Fano and explicitly computing their effective and movable cones for specific ranges of $n$ and the number of points $s$ .

Original authors: Elisa Postinghel, Artie Prendergast-Smith

Published 2026-02-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Elisa Postinghel, Artie Prendergast-Smith

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to understand the shape and stability of a very complex building. In the world of mathematics, this building is a geometric space called a variety. Specifically, this paper looks at a building constructed by taking two flat sheets of space (like a piece of graph paper and a piece of 3D space) and gluing them together, then poking holes in them at specific spots.

The authors, Elisa Postinghel and Artie Prendergast-Smith, are trying to answer three big questions about these buildings:

What does the building look like? (Can we describe its shape completely?)
Is it stable? (In math terms, is it "Log Fano"? This is a fancy way of saying the building has a nice, balanced structure that makes it easy to study.)
What are its limits? (If you try to stretch or shrink the building, what are the boundaries before it breaks?)

Here is a breakdown of their journey, using simple analogies.

1. The Setting: The "Glued Sheets"

Imagine you have a flat sheet of paper ( $P^n$ ) and a slightly larger, 3D sheet of paper ( $P^{n+1}$ ). You glue them together side-by-side. Now, imagine you take a needle and poke $s$ holes through both sheets at random spots. The result is a new, complex shape.

The authors are studying what happens when the number of holes ( $s$ ) is just slightly more than the size of the smaller sheet ( $n$ ).

If you poke too few holes, the shape is simple and predictable (like a standard house).
If you poke too many holes, the shape becomes chaotic and impossible to fully describe (like a house with too many rooms and no walls).
The "sweet spot" they are investigating is when the number of holes is just right to make the shape interesting but still manageable.

2. The New Tool: "Bilinear Secants"

In classical geometry, if you have a curve (like a snake), you can draw lines connecting pairs of points on it. The collection of all those lines forms a "secant variety." Think of it like a spiderweb spun between points on a snake.

The authors invented a new tool called Bilinear Secants.

The Analogy: Imagine the snake is actually a "double snake" made of two intertwined threads (one on the paper sheet, one on the 3D sheet).
Instead of just connecting points with a straight line, they connect them with a "double line" that respects the structure of both sheets simultaneously.
This creates a new kind of web or "join" that exists in the space between the two sheets.

They discovered that for their specific building (with $n+3$ holes), there is one special, unique "double snake" (called the distinguished rational curve) that passes through all the holes. The "webs" (bilinear secants) spun by this snake turn out to be the structural beams of the building.

3. The Discovery: The "Beams" Hold It Together

The main breakthrough of the paper is realizing that these "webs" (bilinear secant varieties) are not just decorations; they are the extremal rays of the building's shape.

The Cone of Possibilities: Imagine the building has a "shadow" cast by the sun. This shadow represents all the possible ways you can stretch or shrink the building. The edges of this shadow are the "extremal rays."
The Finding: In simpler buildings, the edges of the shadow were just the holes (exceptional divisors) or lines pulled from the original sheets. But in these complex buildings, the edges of the shadow are actually these bilinear webs.
Why it matters: If you know the edges of the shadow, you know the entire shape. By identifying these webs as the edges, the authors can map out the entire geometry of the building.

4. The Verdict: Is the Building Stable? (Log Fano)

The authors prove that for certain numbers of holes, these buildings are Log Fano.

The Metaphor: Think of a "Log Fano" building as a perfectly balanced mobile hanging from the ceiling. It's stable, symmetrical, and if you push it, it swings back to its original position.
The Result: They showed that for buildings with up to $n+2$ holes, and even for the tricky case of $n+3$ holes (in dimensions 2 and 3), the structure is stable. This means mathematicians can use powerful tools to study them without the math breaking down.

5. The "Cone Method": Solving the Puzzle

To figure out the exact shape of the shadow (the "Effective Cone"), they used a technique called the Cone Method.

The Analogy: Imagine you are trying to find the boundaries of a room by throwing darts. You throw a dart (a divisor) and see if it hits a wall (a base locus).
They developed a set of rules (Base Locus Lemmas) to predict exactly where the walls are.
They found that the walls are formed by:
1. The holes themselves.
2. Lines pulled from the original sheets.
3. The Bilinear Webs (the new discovery).
4. Some unexpected "pencils" (families of lines) that act like hidden supports.

Summary

This paper is like a master architect discovering that a complex, multi-story building isn't held up by standard pillars, but by a unique, woven net of "double lines" connecting specific points.

By identifying this net (the bilinear secant varieties), they were able to:

Draw the complete blueprint of the building's shape.
Prove the building is stable and well-ordered.
Open the door to understanding even taller, more complex versions of these buildings in the future.

It turns a chaotic, high-dimensional math problem into a structured, solvable puzzle by finding the hidden "spiderwebs" that hold the geometry together.

1. Problem Statement and Context

The paper investigates the birational geometry of varieties $X_{n,m}^s$ , defined as the blow-up of the product of projective spaces $\mathbb{P}^n \times \mathbb{P}^m$ in $s$ general points. The primary goal is to determine when these varieties possess "good" finiteness properties, specifically:

Log Fano: The existence of an effective $\mathbb{Q}$ -divisor $\Delta$ such that $(X, \Delta)$ is klt (Kawamata log terminal) and $-(K_X + \Delta)$ is ample.
Mori Dream Space (MDS): A variety where the Cox ring is finitely generated, implying that the effective and movable cones are rational polyhedral.

Current State of Knowledge:

For $n=m$ (equal dimensions) or $s \le n+1$ (toric cases), the classification of MDS and Log Fano properties is complete.
For $n \neq m$ (unequal dimensions), the problem is significantly harder due to the lack of a Weyl group action that typically simplifies the geometry of equal-dimensional products.
Previous work established results for $X_{1,m}^s$ and $X_{n,n+1}^s$ with $s \le n+1$ .
The Gap: The paper focuses on the case $X_{n,n+1}^s$ where $s = n+2$ and $s = n+3$ . It is known that for $n \ge 5$ and $s \ge n+4$ , these varieties are not MDS. The cases $s=n+2$ and $s=n+3$ were open.

2. Methodology

The authors employ a combination of classical birational geometry techniques and new constructions specific to products of projective spaces:

The Cone Method: Adapted from previous work (GPPS23), this involves computing the effective cone by identifying all necessary "base locus inequalities." If a divisor class fails to satisfy these inequalities, it must contain specific subvarieties in its base locus.
Bilinear Secant Varieties and Joins: The authors introduce a generalization of classical secant varieties to products of projective spaces.
- Bilinear Span ($BL$): The product of linear spans of projections of points onto the two factors.
- Bilinear Join ($BJ$) and Secant ($BS$): Unions of bilinear spans over general points.
- Determinantal Description: They show that bilinear secant varieties of specific rational curves can be described as determinantal varieties (zero loci of minors of specific matrices).
Base Locus Lemmas: The authors derive precise lower bounds for the multiplicity of containment of these special subvarieties (exceptional divisors, pullbacks, bilinear joins) in the base locus of an effective divisor. These lemmas are crucial for defining the facets of the effective cone.
Log Fano Construction: To prove the Log Fano property, they explicitly construct a boundary divisor $\Delta$ $Δ$ (a sum of specific effective divisors with rational coefficients) and verify:
1. $-(K_X + \Delta)$ is ample (by checking intersection with curve generators).
2. The pair $(X, \Delta)$ is klt (by computing discrepancies on a log resolution obtained by blowing up the singular loci of $\Delta$ ).

3. Key Contributions and Results

A. Introduction of Bilinear Secant Varieties

The paper defines bilinear secant varieties $BS_k(X)$ and bilinear joins $BJ(X_1, \dots, X_k)$ for subvarieties of $\mathbb{P}^n \times \mathbb{P}^m$ .

They focus on the distinguished rational curve $C \subset \mathbb{P}^n \times \mathbb{P}^{n+1}$ of bidegree $(n, n+1)$ passing through $n+3$ general points.
They prove $BS_k(C)$ are irreducible determinantal varieties defined by the maximal minors of a specific matrix $M_k$ .

B. Case $s = n+2$ : The Variety $X_{n,n+1}^{n+2}$

Effective and Movable Cones: The authors compute the effective cone $\text{Eff}(X_{n,n+1}^{n+2})$ $Eff (X_{n, n + 1}^{n + 2})$ . The extremal rays are generated by:
1. Exceptional divisors $E_i$ .
2. Pullbacks of fixed hyperplanes from the factors ( $H_1 - \sum E_I$ and $H_2 - \sum E_J$ ).
- Note: Unlike the $s=n+3$ case, no bilinear secant varieties appear as generators here.
Log Fano Property: They prove Theorem 5.4: $X_{n,n+1}^{n+2}$ $X_{n, n + 1}^{n + 2}$ is Log Fano for all $n \ge 1$ $n \geq 1$ .
- They construct an explicit Log Fano pair $(X, \Delta)$ where $\Delta$ is a weighted sum of two divisors pulled back from the factors.
- This implies $X_{n,n+1}^{n+2}$ is a Mori Dream Space.

C. Case $s = n+3$ : The Variety $X_{n,n+1}^{n+3}$

This case introduces new geometric phenomena driven by the distinguished rational curve $C$ .

New Generators: The effective cone now includes generators arising from the bilinear secant varieties of $C$ and their joins with points.
Specific Computations:
- $X_{2,3}^5$ ( $n=2, s=5$ ):
  - The effective cone is generated by exceptional divisors, pullbacks, and the bilinear secant variety $BS_2(C)$ (which is a divisor in this dimension).
  - They prove Theorem 6.6: $X_{2,3}^5$ is Log Fano. The boundary divisor $\Delta$ includes $BS_2(C)$ and other fixed components.
  - They explicitly compute the movable cone.
- $X_{3,4}^6$ ( $n=3, s=6$ ):
  - Here $BS_2(C)$ is not a divisor (it has codimension $>1$ ). However, the bilinear joins of $BS_2(C)$ with points (denoted $D_i$ ) and joins of $C$ with fibers (denoted $Q_i$ ) become the new extremal divisors.
  - They compute the effective cone for $X_{3,4}^6$ (Theorem 7.3), showing it is generated by exceptional divisors, pullbacks, and these new "BSJ" (Bilinear Secant and Join) divisors.

D. Generalization and Conjectures

Proposition 8.1: They propose that for general $n$ , the extremal rays of $\text{Eff}(X_{n,n+1}^{n+3})$ are spanned by "BSJ divisors" (bilinear joins of points and secant varieties of $C$ ), exceptional divisors, and pullbacks.
Open Questions:
- Is $X_{n,n+1}^{n+3}$ Log Fano for all $n$ ? (Confirmed for $n=1, 2$ ; difficult for $n=3$ due to complex log resolutions).
- Which higher-dimensional cases ( $s \ge n+4$ ) are MDS?

4. Significance

Extension of Classical Theory: The paper successfully extends the theory of blow-ups of $\mathbb{P}^n$ (where secant varieties of rational normal curves play a central role) to the product setting $\mathbb{P}^n \times \mathbb{P}^{n+1}$ .
New Geometric Objects: The introduction of bilinear secant varieties provides a new toolkit for studying products of projective spaces, revealing that these varieties play the same structural role in the birational geometry of $X_{n,n+1}^s$ as classical secant varieties do for $X_{n}^s$ .
Resolution of Open Cases: It settles the Log Fano and MDS status for $X_{n,n+1}^{n+2}$ and provides the first complete description of the effective cones for $X_{2,3}^5$ and $X_{3,4}^6$ .
Complexity of Unequal Dimensions: The results highlight that products of unequal dimensions lack the symmetry of equal-dimensional products (no Weyl group action), requiring the discovery of new, dimension-dependent geometric structures (like the specific bilinear joins) to describe their cones.

In summary, the paper establishes that the birational geometry of blow-ups of $\mathbb{P}^n \times \mathbb{P}^{n+1}$ is governed by the interplay between classical pullback divisors and the newly defined bilinear secant varieties of a distinguished rational curve, providing a complete picture for $s \le n+3$ in low dimensions and a framework for higher dimensions.