diff --git a/src/main/scala/sbt/VirtualAxis.scala b/src/main/scala/sbt/VirtualAxis.scala
index 7d83168..ff0b33c 100644
--- a/src/main/scala/sbt/VirtualAxis.scala
+++ b/src/main/scala/sbt/VirtualAxis.scala
@@ -69,6 +69,15 @@ object VirtualAxis {
     compare(sbv1, sbv2) || compare(sbv2, sbv1)
   }
 
+  // This admits partial Scala version
+  private[sbt] def isPartialVersionEquals(ax1: VirtualAxis, ax2: VirtualAxis): Boolean = {
+    (ax1, ax2) match {
+      case (ax1: ScalaVersionAxis, ax2: ScalaVersionAxis) =>
+        (ax1 == ax2) || (ax1.value == ax2.value)
+      case _ => ax1 == ax2
+    }
+  }
+
   case class ScalaVersionAxis(scalaVersion: String, value: String) extends WeakAxis {
     override def idSuffix: String = directorySuffix.replaceAll("""\W+""", "_")
     override val suffixOrder: Int = 100
diff --git a/src/main/scala/sbt/internal/ProjectMatrix.scala b/src/main/scala/sbt/internal/ProjectMatrix.scala
index ddfcc51..d32aced 100644
--- a/src/main/scala/sbt/internal/ProjectMatrix.scala
+++ b/src/main/scala/sbt/internal/ProjectMatrix.scala
@@ -112,6 +112,8 @@ sealed trait ProjectMatrix extends CompositeProject {
   def nativePlatform(scalaVersions: Seq[String], settings: Seq[Setting[_]]): ProjectMatrix
   def native: ProjectFinder
 
+  def defaultAxes(axes: VirtualAxis*): ProjectMatrix
+
   def projectRefs: Seq[ProjectReference]
 
   def filterProjects(axisValues: Seq[VirtualAxis]): Seq[Project]
@@ -193,6 +195,7 @@ object ProjectMatrix {
       val configurations: Seq[Configuration],
       val plugins: Plugins,
       val transforms: Seq[Project => Project],
+      val defAxes: Seq[VirtualAxis],
   ) extends ProjectMatrix { self =>
     lazy val resolvedMappings: ListMap[ProjectRow, Project] = resolveMappings
     private def resolveProjectIds: Map[ProjectRow, String] = {
@@ -200,12 +203,16 @@ object ProjectMatrix {
         r <- rows
       } yield {
         val axes = r.axisValues.sortBy(_.suffixOrder)
+          .filterNot(isSortOfDefaultAxis)
         val idSuffix = axes.map(_.idSuffix).mkString("")
         val childId = self.id + idSuffix
         r -> childId
       }): _*)
     }
 
+    private def isSortOfDefaultAxis(a: VirtualAxis): Boolean =
+      defAxes exists { da => VirtualAxis.isPartialVersionEquals(da, a) }
+
     private def resolveMappings: ListMap[ProjectRow, Project] = {
       val projectIds = resolveProjectIds
 
@@ -240,8 +247,11 @@ object ProjectMatrix {
             name := self.id
           )
           .settings(
-            r.scalaVersionOpt.toList map { sv =>
-              Keys.scalaVersion := sv
+            r.scalaVersionOpt match {
+              case Some(sv) =>
+                List(Keys.scalaVersion := sv)
+              case _ =>
+                List(Keys.autoScalaLibrary := false, Keys.crossPaths := false)
             }
           )
           .settings(
@@ -279,7 +289,8 @@ object ProjectMatrix {
             case (Some(depProjId), Some(depSV), Some(depSBV), Some(depCross)) =>
               if (sbv == depSBV || depCross != CrossVersion.binary)
                 Some(
-                  depProjId.withConfigurations(dep.configuration).withExplicitArtifacts(Vector.empty)
+                  depProjId.withConfigurations(dep.configuration)
+                    .withExplicitArtifacts(Vector.empty)
                 )
               else if (VirtualAxis.isScala2Scala3Sandwich(sbv, depSBV) && depCross == CrossVersion.binary)
                 Some(
@@ -288,7 +299,7 @@ object ProjectMatrix {
                     .withConfigurations(dep.configuration)
                     .withExplicitArtifacts(Vector.empty)
                 )
-              else sys.error(s"scalaBinaryVersion mismatch: expected $sbv but found ${depSBV}")
+              else sys.error(s"scalaBinaryVersion mismatch: expected $sbv but found ${depSBV} in $depProjId")
             case _ => None
           }
         }
@@ -382,6 +393,9 @@ object ProjectMatrix {
             .settings(settings)
         })
 
+    override def defaultAxes(axes: VirtualAxis*): ProjectMatrix =
+      copy(defAxes = axes.toSeq)
+
     def scalajsPlugin(classLoader: ClassLoader): Try[AutoPlugin] = {
       import sbtprojectmatrix.ReflectionUtil._
       withContextClassloader(classLoader) { loader =>
@@ -491,6 +505,7 @@ object ProjectMatrix {
         configurations: Seq[Configuration] = configurations,
         plugins: Plugins = plugins,
         transforms: Seq[Project => Project] = transforms,
+        defAxes: Seq[VirtualAxis] = defAxes,
     ): ProjectMatrix = {
       val matrix = unresolved(
         id,
@@ -502,7 +517,8 @@ object ProjectMatrix {
         settings,
         configurations,
         plugins,
-        transforms
+        transforms,
+        defAxes,
       )
       allMatrices(id) = matrix
       matrix
@@ -511,7 +527,8 @@ object ProjectMatrix {
 
   // called by macro
   def apply(id: String, base: sbt.File): ProjectMatrix = {
-    val matrix = unresolved(id, base, Nil, Nil, Nil, Nil, Nil, Nil, Plugins.Empty, Nil)
+    val defaultDefAxes = Seq(VirtualAxis.jvm, VirtualAxis.scalaPartialVersion("2.13.3"))
+    val matrix = unresolved(id, base, Nil, Nil, Nil, Nil, Nil, Nil, Plugins.Empty, Nil, defaultDefAxes)
     allMatrices(id) = matrix
     matrix
   }
@@ -526,7 +543,8 @@ object ProjectMatrix {
       settings: Seq[Def.Setting[_]],
       configurations: Seq[Configuration],
       plugins: Plugins,
-      transforms: Seq[Project => Project]
+      transforms: Seq[Project => Project],
+      defAxes: Seq[VirtualAxis],
   ): ProjectMatrix =
     new ProjectMatrixDef(
       id,
@@ -538,7 +556,8 @@ object ProjectMatrix {
       settings,
       configurations,
       plugins,
-      transforms
+      transforms,
+      defAxes,
     )
 
   def lookupMatrix(local: LocalProjectMatrix): ProjectMatrix = {
diff --git a/src/sbt-test/projectMatrix/jvm-sandwich/test b/src/sbt-test/projectMatrix/jvm-sandwich/test
index 62f41e9..7625926 100644
--- a/src/sbt-test/projectMatrix/jvm-sandwich/test
+++ b/src/sbt-test/projectMatrix/jvm-sandwich/test
@@ -1,5 +1,5 @@
-> fooAppJVM0_23/compile
+> fooApp0_23/compile
 
-> barAppJVM2_13/compile
+> barApp/compile
 
-> bazAppJVM2_13/check
+> bazApp/check
diff --git a/src/sbt-test/projectMatrix/jvm-with-scoping/build.sbt b/src/sbt-test/projectMatrix/jvm-with-scoping/build.sbt
index a7cbf20..998141a 100644
--- a/src/sbt-test/projectMatrix/jvm-with-scoping/build.sbt
+++ b/src/sbt-test/projectMatrix/jvm-with-scoping/build.sbt
@@ -1,3 +1,5 @@
+lazy val scala213 = "2.13.3"
+lazy val scala212 = "2.12.12"
 lazy val check = taskKey[Unit]("")
 
 lazy val root = (project in file("."))
@@ -11,7 +13,7 @@ lazy val app = (projectMatrix in file("app"))
   .settings(
     name := "app"
   )
-  .jvmPlatform(scalaVersions = Seq("2.12.8"))
+  .jvmPlatform(scalaVersions = Seq(scala213))
 
 lazy val core = (projectMatrix in file("core"))
   .settings(
@@ -22,7 +24,7 @@ lazy val core = (projectMatrix in file("core"))
       assert(directs.size == 2, s"$directs")
     },
   )
-  .jvmPlatform(scalaVersions = Seq("2.12.8", "2.11.12"))
+  .jvmPlatform(scalaVersions = Seq(scala213, scala212))
   .configure(addStuff)
 
 lazy val intf = (projectMatrix in file("intf"))
@@ -33,7 +35,7 @@ lazy val intf = (projectMatrix in file("intf"))
   )
   .jvmPlatform(autoScalaLibrary = false)
 
-lazy val core212 = core.jvm("2.12.8")
+lazy val core213 = core.jvm(scala213)
 
 def addStuff(p: Project): Project = {
   p.settings(
diff --git a/src/sbt-test/projectMatrix/jvm-with-scoping/test b/src/sbt-test/projectMatrix/jvm-with-scoping/test
index a0ccaf5..4a2cee2 100644
--- a/src/sbt-test/projectMatrix/jvm-with-scoping/test
+++ b/src/sbt-test/projectMatrix/jvm-with-scoping/test
@@ -1,6 +1,6 @@
 > compile
 
+$ exists core/target/jvm-2.13/classes/a/Core.class
 $ exists core/target/jvm-2.12/classes/a/Core.class
-$ exists core/target/jvm-2.11/classes/a/Core.class
 
-> coreJVM2_12/check
+> core/check
diff --git a/src/sbt-test/projectMatrix/jvm/build.sbt b/src/sbt-test/projectMatrix/jvm/build.sbt
index e73f056..b7d58fb 100644
--- a/src/sbt-test/projectMatrix/jvm/build.sbt
+++ b/src/sbt-test/projectMatrix/jvm/build.sbt
@@ -1,3 +1,5 @@
+lazy val scala213 = "2.13.3"
+lazy val scala212 = "2.12.12"
 lazy val check = taskKey[Unit]("")
 
 lazy val root = (project in file("."))
@@ -11,7 +13,7 @@ lazy val app = (projectMatrix in file("app"))
   .settings(
     name := "app"
   )
-  .jvmPlatform(scalaVersions = Seq("2.12.8"))
+  .jvmPlatform(scalaVersions = Seq(scala213))
 
 lazy val core = (projectMatrix in file("core"))
   .settings(
@@ -19,7 +21,7 @@ lazy val core = (projectMatrix in file("core"))
       assert(moduleName.value == "core", s"moduleName is ${moduleName.value}")
     },
   )
-  .jvmPlatform(scalaVersions = Seq("2.12.8", "2.11.12"))
+  .jvmPlatform(scalaVersions = Seq(scala213, scala212))
 
 lazy val intf = (projectMatrix in file("intf"))
   .settings(
@@ -29,4 +31,4 @@ lazy val intf = (projectMatrix in file("intf"))
   )
   .jvmPlatform(autoScalaLibrary = false)
 
-lazy val core212 = core.jvm("2.12.8")
+lazy val core213 = core.jvm(scala213)
diff --git a/src/sbt-test/projectMatrix/jvm/test b/src/sbt-test/projectMatrix/jvm/test
index a0ccaf5..4a2cee2 100644
--- a/src/sbt-test/projectMatrix/jvm/test
+++ b/src/sbt-test/projectMatrix/jvm/test
@@ -1,6 +1,6 @@
 > compile
 
+$ exists core/target/jvm-2.13/classes/a/Core.class
 $ exists core/target/jvm-2.12/classes/a/Core.class
-$ exists core/target/jvm-2.11/classes/a/Core.class
 
-> coreJVM2_12/check
+> core/check